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EUCLIDIS 



OPERA OMNIA. 



EDIDERUNT 



L L. HEIBERO ET H. MENOE. 



uoL. vn. 




LIPSIAE 

IV AEDIBUS B. G. TEUBNERI. 
MDCCCXCV. 



EUCLIDIS OPTICA, 
OPTICORUM RECENSIO THE0NI8, 

CATOPTRICA, 

CUM SCHOLIIS ANTIQUIS. 



EDIDIT 



L L. HELBERO, 

FBOrSSSOB DB. PHIL. 



* • • I • 







« ■ r 
' •* ■ J * • • « 



• • * • . 1 1 



•,•( .•• * '•■ • 

LIPSIAE 

IN AEDIBUS B. G. TEUBNEBI. 
MDCCCXCV. 







1 36391 



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LIPtlAB: TTPIB B. O. TXiniVIEt. 



PRAEPATIO. 

Godicibus in hoc uolumine usus suin his: 

L In Opticis genuinis: 

V = cod. Dindobonensis XXXI^ 13 (= philos. Gr. 103 
Lambecius), s. XII, in hac parte bombycinus^ de 
quo u. uol. V p. XXIX sq. hinc Optiea genuina 
primus edidi a. 1882 (Litterargesch. Studien iiber 
Euklid p. 93 — 129), postea locos dubios rursus 
inspexi Uindobonae 1883. 

B = cod. Bodleianus Auct. F 6, 23, bombycinus 
«. XIIL continet f. 1—265' Element. I— XIII, 
f. 265^ initium libri XIV, sed deletum, f. 266—273 
Optica ad p. 60, 17 alia manu, sed eiusdem tem- 
poris. haec pars codicis pessime habita est, ita 
ut multa legi nequeant; quare de eius scriptura 
nihil adfirmatum uolo, nisi quod diserte adnotaui. 
contuli ipse OxoniL 

V = cod. Uatic. 6r. 1038, membr. s. XIII, de qua 
u. uol. V p. V— VL contuli ipse. 

Vat. = cod. Uatic. Gr. 1316, ex libris Fuluii Ursini, 
duobus uoluminibus constans. continet foL 1 — 331 
(bomb. s. XIV) Iliadem cum paraphrasi, f. 332' 
aX^BQXov nim) Ha^aiav aQxovtog xtfjiiaj £ 332^^ 
—336 uacant, f. 337—352 (membx. ^.^^^) ^^vC^Ra. 
ad p. 118, 24 a manu recftiAiiaBVECka» ^^%K». xa.. '^ 



VI PRAEFATIO. 

correcta, f. 353—354 (chart.) Archimedis tceqI 
tcjv vdatc i(pi6ta[iBVG)v. contuK ipse. 
Vat.^ = cod. Uatic. Gr. 1039, chartac. s. XV. continet 
Element. libb. XIV— XV, Optica adj». j^l8, 10, 
Fhaenomena (des. Heat jtSQcg)iQ£cat at Ss s^, Xeycoy 
Srt). locos quosdam inspexi. 

m = cod. Marcian. Gr. 303, bomb. s. XIV, de quo 
u. V p. Vni. descripsi Uenetiis 1881. 

A = cod. Ambros. A 92 sup. quinque fragmenta 
codicum diuersorum*), quorum ultimum (fol. 139 
— 142, chart. s. XV) quattuor foliis formae 
minimae Opticorum continet p. 78, 11 7CQ6g — 
82, 14, p. 86, 9 dQd^dg — 90, 3 idv, p. 104, 16 
&Qa — 110, 3, p. 112, 5 dxtivcDv — 116, 2 tfig. 
i contulit Henricus Menge. 

D = cod. Dresd. lat. Db 86, membr. s. XIV, de 
quo u. Curtze, Zeitschr. f. Math. u. Phys. XXVHI 
hist. Abth. p. 1 sqq. hinc (fol. 111 — 122') sumpsi 
interpretationem Latinam, in qua edenda ortho- 
graphiam codicis neglexi, nisi in uocabulis Graecis, 



*) Cfr. Rivola, Vita di Federico Borromeo p. 314: Hebbe 
cotar apparato suo principio da ima colletta di libri, ch' esso 
Federico dimorando in Roma mosso da magnanimo spirito fece 
nelle pubbliche piazze raccogliere e comperare. Venuto era 
alle orecchie di lui, che molti libri cosi stampati come manu- 
scritti si esponevano tratto tratto dalla rozza ed ignorante 
plebe in pubblico sopra le panche o tavole per esser come 
poco buoni a qualsivoglia bottegaio per invoglio o per altro 
servigio di sua mercatantia venduti, e spiacquegli si fattamente 
r intendere, che a si misera ed infelice sorte si soggettassero 
que' parti .... che . . . ordinb ad un suo familiare, che la citta 
tutta di quando in quando per suo diporto scorresse e cotali 

Hbri, non ostante che per antichita guasti fossero , com- 

perasse ed a casa gli facesse . . . portare. 



PRAEFATIO. VU 

errores uero plerosque retinui; ne quid utilitati 
interpretationis ad Graecum eius fundamentum re- 
stituendum detraheretur; errores, qui ad codicem 
Graecum referri non possunt, plerumque in ad- 
notatione, raro in textu emendaui. descripsi ipse. 

L = cod. Musei Britannici Add. 17,368. inter aKa 
mathematica et astronomica fol. 60 — 69' eandem 
interpretationem habet. locos nonnuUos inspexi. 

Jf = cod. Marcianus lat. 332 s. XIII. inspexi. 
De ratione horum codicum u. Prolegom. I. 

II. Scholia in Optica genuina e solo fere V 
desumpsi (nr. 54 etiam in Vat.^, nr. 72 et 78 etiam 
in A, nr. 89 e solo A), ubi manibus V* V^ V^ V*, 
de quibus u. V p. XI — XII, neglegenter scripta sunt. 
nonnuUa in codice deleta uel erasa sunt; minora 
quaedam, quae satis certo legi non poterant, omisi. 
dubitari non potest, quin omnia scholia ab ipsis 
librariis codicis profecta sint; quare saeculo XII anti- 
quiora non sunt. 

ni. In Opticorum recensione Theonis: 

V = cod. Uatic. Gr. 204, membr. s. X, de quo u. V 
p. XII. Optica habet fol. 42^* — 58' manu recentiore 
(V m. rec.) correcta. 

V = cod. Uatic. Gr. 191, bomb. s. XIII — XIV, de quo 
u. Parthey, Monatsberichte der Berliner Academie 
1863 p. 374 sq. 

p = cod. Paris. 6r. 2390, bombyc. s. XIII; u. Omont, 
Inventaire n p. 251. Optica habet fol. 265-— 275. 
omnes ipse contnli. 



Vm PRAEPATIO. 

IV. Scholia in Opticorum recensionem 
Theonis e multis eodicibos descripsi; ubi V uel alius 
codex antiquior aderat^ iuniores inspexi tantum^ non 
contuli, quod significaui siglo codicis non collati uncis 
incluso. 

V 0=* cod. Uatic. 204; u. supra. 
V^ «s» eiusdem manus recentior (V man. rec). 
V^ «» eiusdem manus recentissima. 
v^ = cod. Uatic. 191 manus recens (a manu 1 nulla 
scholia sunt). 
Vat. = cod. Uatic. 192, bomb. s. XIV (u. Om Scho- 

Keme til Euklids Elementer p. 34). 
Vat. m. 2 = eiusdem manus recentior. 
Vat/ = eiusdem manus recentissima. 

R _ cod. Uatic. 202, chart s. XIV-XV (u. Om 

Scholieme til Euklids Elementer p. 34).*) 
= cod. Ottobon. Gr. 102, chart.s.XVI, foL8-22. 
F =« cod. Laurentianus XXVTII, 10, chart. s. XV. 
A «= cod. Ambros. A 101 sup., chart. s. XV. 
M = cod. Marcianus 304, chart. s. \Y. 
M^ = eiusdem manus recens. 
p = cod. Paris. Gr. 2107, chart. s. XV. 
q = cod. Paris. Gr. 2342, chart. s. XIV. 
r = cod. Paris. Gr. 2350, chart. s. XVI. 
s =^ cod. Paris. Gr. 2351, chart. s. XVI. 
t = cod. Paris. Gr. 2363, chart. s. XV. 
u = cod. Paris. Gr. 2472, chart. s. XTV. 
X = cod. Paris. Gr. 2390, de quo u. supra (= p). 

''') Scholia nr. 13 et 14 p. 254—256 (E^) errore hic ppsita 
sunt; pertment ad Optica antiqua, ubi inter scholia sunt nr. 6 
et 8, et petita sunt e cod. Uatic. 10S9 («« Vat.*). 



PRAEFATIO. IX 

V. In Catoptricis usus sum his (ipse contuli): 

V = cod. Uatic. Gr. 204 fol. 135 — 144% de quo u. 
supra; correctus est initio manu recentissima. 

V = cod. Uatic. Gr. 191; u. supra. 
M = cod. Marcianus 303; u. supra. 

m = cod. Marcianus 301, chart. s. XV. 

VI. Scholia Catoptricorum sumpsi ex his: 

V = cod. Uatic. Gr. 204. 

V^ = eiusdem manus recens; u. supra. 

p = cod. Paris. Gr. 2107; u. supra. 

q = cod. Paris. Gr. 2342; u. supra. 
q^ = eiusdem manus eadem atramento rubro. 
p. 14, 2 pro l'6rj scribendum Hea, 

Scr. Hauniae mense Nouembri MDCCCXCIV. 

I. L. Heiberg. 



I. 

De eodieibitft Oplicoruxii gemiiiiarum. 

Codicum supra enumeratoram duae classes distinguuntur, 
YVat.^m et BVat.r, quarum principes sunt VB, et ita prin- 
cipes, mt eeteris niliil sit momenti. nam primum Vat.y non 
modo nusquam meHora praebent quam B, sed etiam in errori- 
bus stoltissimis cum eo consentinnt, uelut p. 14, 13 (o^ ortnm 
e eompendio Q^); 1&, 16; 62, 34. et eos ex ipso B originem 
docere ostendusl loci, ubi compendia codicis B errores genue- 
nmt^ uelut p. 42, 13; 46, 25, 26 — in B enim saepe scribitur 
-♦" pro -^otfccv —; 54, 2 ; 56, 3 ; cfr. praeterea de Vat. p. 32, 25 
34, 24; 50, 16; 52, 11, de Y p. 30, 5; 32, 24; 46, 14 (|3«4»7]^) 
52, 10, 13; 56, 11, 13, 19*), et p. 20, 9, 10, 15; 22, 1; 38, 14, 19 
42, 3 al., ubi in y legitur v pro y, quia in B hae litterae di- 
stingui uiz possunt; etiam p. 20, 16 077 pro stj in y legitur, 
quia € littera in B obscurius scripta est. loco, qui est 
p. 30, 17, cogimnr codicem inter B et Vat.y intermedium sta- 
taere; nam yerba t^ Sijjsi rjUov m B sunt. huic classi ad- 
cedit A (u. p. 80, 12, 16, 20; 82, 8, 12; 88, 5, 7, 11, 20; 90, 1, 2; 
104, 17, 21, 23, 25; 106, 3, 4, 8; 108, 6; 114, 12; 116, 1). Vat. 
a mann 2 ad similitudinem codicis V correctus est (p. 22, 19; 
26, 2; 28, 14; 74, 7; 88, 6 al.), postquam is a manu 2 correctus 
erat (p. 4, 27; 24, 4; 28, 7; 36, 25; 40, 9; 46, 14; 60, 7; 82, 26). 

in altera classe Vat.^m ex eodem archel^o deriuatos esse, 
adparet ex summo eorum in erroribus consensu (p. 2, 7, 8 ; 8, 1 ; 
24, 14; 34, 12; 36, 4, 10; 68, 13, 16; 76, 16; 78, 14, 16; 106, 2; 
108, 16; 116, 18); nam Vat.^ ex m descriptum non esse, ex 
p. 68, 21 et p. 108, 11 (ytvofuivoDv) concludi potest. nec dubium 
est, quin hic archetypus ex V pendeat; nam quae m meKora 
habet (p. 10, 10, 25; 102, 19; 104, 6 al.) — e Vat.^ nihil eius- 

♦) Hbc rcferri potest etiam p. 98, 3, libi B «^Lfc ^>3Nsv5k*-v^^ 
compendimn habuit, quod Vat. 



XIV PROLEGOMEKA. 

modi enotaui — , librario debentur, qui alia quoque suo ar- 
bitrio emendauit, uelut p. 24, 9; 68, 17, 20; 74, 4, 9, 10, 11; 
80, 11; 82, 21, 22; 88, 7; 92, 9, quibus locis consensus alterius 
classis cum V interpolationem arguit. et est, cur putemus, 
hunc archetypum communem esse cod. Laurent. XX^VIII, 6 (f ), 
quem e V descriptum esse demonstraui V p. XXVI sq.; cfr. 

p. 4, 8 7t(XQa(pSQ0niv(av] V, nq)SQOiisv(av f, TCSQKpSQOiisvGiv m; 

p. 10, 26 nX] V, 2 f , x^ Vat.^m; p. 12, 21 JFK] in ras. V, 
py% fm; p. 50, 7 r&v yimvcov] V, rov %vXh9Q0V mg. m. 2; r&v xo»- 
vcav f, yLvXlvdQCDv mg. m. 1 ; t&v %Sv(av xal r&v tivXlvdQav Vat.^m; 
p. 68, 8 AFZ] corr. ex ^4 V, aj"f, ^a^ Vat.^ /Jaf m; p. 82, 26 

BJr] p9y V (h. e. Bz/ZT), /Jfdy fVat.^m; p. 102, 17 ra] V, 
Tcc yoQ f, roc ydcQ deleto ydQ Vat.*; p. 108, 5 inl fiL&g — 7 F, Z, A] 
mg. m. 2 V, mg. m. 1 paullo superius f, mg. m. 1 ad p. 106, 26 
Vat.^, om. m; p. 108, 13 nQOTiyslG^ocL] V, TCQOTista&at. fVat.^m.*) 

praeter codices iam commemoratos etiam cod. Laurent. 
XXVni, 3 Optica nostra habet, sed in hac parte (qp) ex f de- 
scriptus est, ut demonstraui V p. XXVI. in cod. Uatic. Gr. 246 
(chart. s. XV) inter alia fol. 17' leguntur definitiones his scripturis 
uariantibus: p. 2, 1 Ev%Xsi8ov 6nxi%ol oqoi, 8 &v om., nQoa- 
supra scr., 11 $s] 9\ ildc66ovos iXdcaeova, 16 iikv ^no] inl. 
alios codices non inueni. 

restant igitur soli VB; quorum V praeferendus esse uidetur, 
non modo quod antiquior est, sed etiam quod B, quamquam 
saepe meliorem scripturam habet (p. 12, 24—25; 16, 17; 20, 1; 
22, 19; 24, 9; 34, 23; 36, 14; 52, 2; 54, 13; 56, 20; item Vat.v, 
ubi B deest, p. 60, 18; 68, 2; 74, 7; 82, 5; 88, 6; 100, 24; 114, 15 
— scripturas p. 16, 17; 36, 14; 74, 7 confirmat recensio Theonis 
p. 166, 14; 180, 9; 210, 12 — ; fortasse etiam p. 8, 12; 26, 20; 
34, 21; 38, 1; 42, 16 et p. 40, 9; 66, 22 hi Ss, quae confirmantur 
scriptura Theonis p. 184, 4; 204,14), ab interpolationis suspicione 
liber non est. uelut p. 6, 26 erroris origo intellegitur e scrip- 
tura codicis V, e scriptura codicis B non intellegitur; p. 18, 10; 
36, 16 error non recte correctus est, p. 26, 11 cum V m. 2 in 
coniectura superflua conspirat, item p. 20, 28 (cfr. enim 



*) P. 120, 6 haec est scriptura codicis f: arm. Srt ovtcog 
dcpslXsL yQcc(psivoci ('fjvai)' iccv Ss i} inl t^v avvatpriv xr\v did- 
fistQOv (scr. t&v 8iaiLstQ(ov) nrjts TtQbg dQd^as iitl (scr. 17) tA 
iTtLTtidcp cetera ut V, qui hinc restitui potest; cfr. m. 



PROLEGOMENA. XV 

p. 22, 1, 15); cfr. praeterea p. 6, 27; 30, 16; 40, 18. similiter, 
ubi B deest, Vat.v cum V male correcto conspirant p. 70, 4, 
coniecturam falsam habent p. 80, 12; cfr. p. 64, 11. praeterea 
error p. 118, 21 in Vat. ex eo compendio ortus est, quod V 
seruauit. p. 84, 18 nunc dubito, an dQ&rjv cnm V omittendum 
sit, quamquam apud Theonem p. 216, 12 exstat, sed ante 
yaviav; nam p. 36, 18; 38, 23 B ad similitudinem recensionis 
Theoninae p. 180, 16; 182, 17 postea correctus est et p. 64, 4 
cum ea p. 192, 19 in scriptura minus exquisita contra V con- 
spirat. etiam recensio Theonis interpolationes codicis B arguit 
p. 162, 3 (= p. 10, 19); 170, 2 (= 20, 4); 180, 14 (== 36, 16); 
188, 18 (= 46, 13); 196, 22 (= 58, 15); 202, 6 sq. (= 64, 11); 
204, 14 (= 66, 22 svd^ela om.); 244, 3 (= 118, 4). itaque in 
recipiendis scripturis classis secundae, etiam si per se bonae 
sunt, caute agendum, nec sine certa causa a V discedendum. *) 
Interpretationem Latinam e D solo edidi, quia eum solum 
totumconferrepotui; sine dubioaliunde emendari potest. speci- 
minis causa huc congeram, quae notaui (praeter pauca, quae in 
adparatu dedi) ex ML et cod. Amplon. Q 387 (saec. XIV, fol. 47' 
— 52', in fine: explicit liber de uisibus). p. 3, 1 rectas ductas] 
eductas rectas Ampl.; ductas lineas] lineas eductas ML; 
2 inmensarum] inmensurarum L in ras.i uisibus] uisibus qui- 
dem MLAmpl. (cfr. p. 2, 3); 5 inciderit] incidunt L; 6 inciderit] 
incidunt L; 7 quidem] om. M; uero minori] minori uero ML; 

11 quidem] om. M; 16 enim] quidem Ampl.; quidem] owk 
Ampl.; esto] om. M; 17 incidunt Ampl.; igitur] ergo Ampl.; 
p. 5, 1 uisus incidentes M; 2 fient] fieret L; et] puncta M 
in ras.; 3 non ergo uidebitur] mg. m. 1 Ampl.; simul uide- 
bitur M; 4: ad totum M; simul] om. L; 8 quidem oculus L; 
10 sit] om. L; 19 enim] om. L; autem] om. L; p. 7, 1 iam 
non L; 8 e] a Ampl.; minora] maiora Ampl.; p. 121, 8 epi- 
pedo Ampl.; 9 diametrorum] om.Ampl.; 11 nec] neque Ampl.; 

12 demonstrabuntur jlwpZ. pro p. 5, 12 trigoni — 16 uidentur 
ci p. 7, 2 ad fc — 4 uidebitur prorsus dlia habet L; et omnino 
etiam in aliis codicibus aliae exstant a Graecis uerbis difFe- 



*) P. 72, 7 cum V correcto et B pro yocQ scribendum esse 
olv, ostendit recensio Theonis p. 208, 16; p. 68, 16 pro &ats 
bUsIv supplendum %ai eIglv cum Theone p. 204, 26 ; itaque classia 
secunda uestigium ueri seruauit. p. 16, ^l -vial i.^^Ti.^Ta.., ojcs^s^ 
in correcto demum V additur, cuius nxiWa. fesX» ^■vvR.\>wXaa. 



XVI PROLEGOMENA. 

rentiae, de quibns in eap. m locns erit dicendi. si his inter- 
polationibn& ad tempus omissis interpretationem cum Graecis 
codicibns comparamus, adparet, eam eodioe Graeco niti, si 
summam spectes, nostris simili; iidem enim errores occnrmnt, 
uelnt p. 6, 18; 7, 6; la, 11; 15, 6; 17, % 9; 19, 4—5; 26, », 5; 
67, 16; 69, 13; 81, 11; 83, 4; 89, 6; 96, 1; 97, 6, 17; 108, 7; 
107, 8, 4 (cfir. p. 106^ 16); 109, 7; 115, 3—4. scripturam emen- 
datiorem raro habuit, uelnt p. 29, 11; 115, 2; 117, 14, et for- 
tasse p. 81, 2 (ng — XsyBtai p. 30, 8 om.); p. 81, 6 (t&v ii 
— nUvQal p. 80, 7—8 om.); p. 71, 28 (fort. scr. Slcc fv 
%iVT(fOv ai AB, Fd p. 70, 19); p. 81, 13 {n&cai &viaoi 
p. 80, 17?); p. 89, 8 (pro FEA p. 88, 8 fort. FEH); p. 97, 20 
(pro ^E p. 96, 23 melius EA). lacunae p. 37, 2; 39, 7; 77, 12; 
79, 16 ; 87, 21 ; 99, 19 fortasse librario debentnr. raro enm Y 
conspirat (p. 11, 12; 19, 8; 27, 15; 31, 6, 14; 39, 5; 47, 11; 
49, 6; 71, 1; 83, 9; 89, 9?), contra cum altera classe summus 
est COnsensns (p. 7, 16, 18; 9, 11,24; 11, 12, 19; 13, 12; 15, 21; 
17, 13, 20; 19, 4, 6, 17; 21, 22; 25, 7; 27, 9, 14; 83, 3; 35, 14; 
37, 6, 7, 15; 39, 3, 16; 41, 2, 4, 7; 43, 12; 45, 3; 47, 7, 8; 53, 
6, 12; 56, 12; 57, 11; 63, 2, 17,18, 21; 66, 6, 6, 8, 10, 11, 12,15; 
67, 3, 6, 7, 15; 69, 2, 3, 8; 71, 10; 75, 4; 79, 6; 88, 15; 85, 5; 
89, 7, 9; 91, 12, 14; 97, 15, 16; 101, 19; 103, 9; cfr. p. 25, 16; 
36, 16; 41, 1; 103, 17; cum V solo p. 105, 7, cxim Vat solo 
p. 61, 6X etiam in mendis apertis (p. 71, 16; 73, 6; 75, 9; 79, 4; 
101, 6; 105, 15; 107, 3, 4; 115, 8; cfr. p. 5, 13; 15, 8; 97, 14). 
in hac tanta constantia memorabile est, eam locis haud ita 
paucis etiam cum m e priore classe consentire (p. 27, 14; 37, 11 ; 
67,18; 69,12; 75,7; 79,4), interdum in erroribus grauioribus, 
uelut p. 37, 13; 83, 8; 99, 10, 22. de origine eius in cap. UI 
uidebimus. 

n. 

De codicibus Opticorum Theoninonim. 

Primum codices praeter Vv, qui et Optica Theonis et 
Catoptrica continent, enumeremus. 

1) cod. Uatic. Gr. 192 s. XIV, u. supra. 

2) cod. Ottobon. Gr. 102, chartac. s. XVI, ex codicibus 
lohannis Angeli Ducis ab Altaemps. continet Catoptrica, Optica, 
Heliodorum, Arrianum in Epictet. 

3) cod. Angelic. C 2, 9, chartac. s. XV; u. Om Scholieme 
til Euklids Elementer p. 34. 



PROLEGOMENA. XVH 

4) cod. Scorial. X — I — 4, chartac. s. XVI; scripsit Ualerianus 
Foroliniensis. continet Catoptrica, Fhaenomena, Optica cum 
sclioliis. 

5) cod. Paris. Gr. 2107, chartac. s. XIV--XV. continet 
inter uaria mathematica, astronomica, medica (u. Omont II 
p. 196) fol. 27—68 Optica et Catoptrica. 

6) cod. Paris. Gr. 2342, chartac. s. XTV, u. Apollon. II p. XII 
et LXIX. Optica habet fol. 109—113, Catoptrica fol. 116—118'. 

7) cod. Paris. Gr. 2347, chartac. s. XVI. continet Elem. 
I — Xin, Data, Marinum, Optica fol. 346 — 364, Catoptrica 
fol. 365 — 376, Hypsiclem, Phaenomena. 

8) cod. Paris. Gr. 2350, chartac. s. XVI; scripsit Petrus 
Uergetius. u. Om Scholieme til Euklids Elem. p. 56. 

9) cod. Paris. Gr. 2352, chartac, scr. lohannes Ehosus a. 1487 
— 1488. continet Proclum in Elem., Catoptrica, Phaenomena, 
Optica, Data. 

10) cod. Paris. Gr. 2366, chartac. s. XVI; scripsit lohannes 
Hydruntinus. u. Om Schol. t. Eukl. Elem. p. 34. 

11) cod. Paris. Gr. 2468, chartac, scr. Angelus Uergetius 
a. 1565. continet Optica, Catoptrica, Phaenomena. 

12) cod. Paris. suppl. Gr. 186, chartac, scr. Angelus Uer- 
getius a. 1637. continet Elem. I— XV, Catoptrica, Optica. 

13) cod. Paris. suppl. Gr. 195, chartac. s. XV. continet 
Catoptrica, Optica, Anonymi Optica, Isagog. harmon. fuit 
'AXpi/j^ov 'Pvft^oxrtov xal r&v anovdaiav; f. 1 mg.: 1507 Uenetiis 
And. Conerj. 

14) cod. Monac Gr. 361, bomb. s. XIII. continet praeter 
Optica fol. 8 — 14 et Catoptrica fol. 15 — 17' sine ordine Phaeno- 
mena, Data, Ptolemaei Harmon. 

15) cod. Berolin. Philipps. Gr. 1542, chartac s. XVI. con- 
tinet Catoptrica, Phaenomena, Optica, Data. 

16) cod. Oxon. coll. S. lohannis 55, chartac. s. XVI. continet 
Optica, Phaenomena, Catoptrica („ex dono Reuerend. in Christo 
Patris Gul. Laud, Archiepiscopi Cantuariensis Anno 1642"). 

17) cod. Cantabrig. Uniuersit. Gg 11, 33, chartac s. XV 
— ^XVI. continet inter multa alia mathematica et astronomica 

.(Coxe in p. 58 sq.) Optica fol. 248—251', 252—253, 107—109' 
(propp. 1 — 24 cum scholiis) et Catoptrica fol. 258 — 261. 

18) cod. Cantabrig. Uniuersit. Nn in, 8, chartac s. XVI. 
continet Catoptrica, Phaenomena, Optica. 

19) cod. Bodleian. Baroccian. 161, c\i«i.TW.. ^.'^ . ^«^Tii^^ 
Euclides, edd. Heiberg et Menge. "VH. ^ 



XVm PROLEGOMENA. 

fol. 196 — 380 Catoptrica, Phaenoinena, Optica, Data. praece- 
dunt et sequuntur alia mathematica, u. Coxe I p. 276. 

20) cod. Leidensis 7, chartac. s. XVI, de quo u. V p. CIV. 

21) cod. Barberin. n, 81, chartac. s. XV. continet Cat- 
optrica fol. 1—7, 32—35', Optica fol. 59«— 80', praeterea sine 
ordine Phaenomena, Data, commentarium in Cleomedem, Heronis 
G«odaesiam. 

22) cod. Ambros. A 101 sup., de quo u. ApoUon. n p. Xn, 
nisi quod nunc adfirmare possum, codicem chartaceum esse 
saec. XV— XVI. 

23) cod. Uindobon. suppl. 9, chartac. s. XVH; u. Apollon. 11 

p. xm. 

24) cod. Uindobon. Grr. 120 praeter mechanica quaedam 
fol. 37 — 39' fragmenta habet Catoptricorum (definitiones, 
propp. 1, 3, 4, ult.) et Opticorum (propp. 18 — 21). 

25) cod. Toletan. Biblioth. Capitul. 98 — 13, chartac. s. XVI; 
u. Graux et Martin, Notices p. 278. continet Catoptrica, Phaeno- 
mena, Optica, Data. 

Optica sola sine Catoptricis hi codices habent: 

26) cod. Uatic. Gr. 202, chartac. s. XIV— XV, u. supra p. Vm. 

27) cod. Laurent. XXVm, 10, chartac. s. XV. continet Data, 
Optica, Phaenomena. 

28) cod. Marcian. 304, chartac. s. XV. continet Optica, 
Autolycum de sphaera mota, Theodosium de habitat., de diebus, 
Aristarchum, Autolycum de ortu, Hypsiclem. 

29) cod. Paris. Gr. 2351, chartac. s. XVI; scripsit Constan- 
tinus Palaeocappa. continet Phaenomena et fol. 65 — 116 Optica. 

30) cod. Paris. Gr. 2363, chartac. s. XV. inter alia mathe- 
matica et astronomica (u. Omont H p. 246 — 247)*) Optica habet 
fol. 29^—40'. 

31) cod. Paris. Gr. 2390, bomb. s. Xm, =« p; u. supra. 

32) cod. Paris. Gr. 2472, chartac. s. XTV. inter alia mathe- 
matica et astronomica (u. Omont H p. 266 — 267) Optica con- 
tinet foL 49—63. 

De aestimatione horum codicum nunc aliter iudico, ac 
cum ante hos quinque annos textum huius uoluminis recense- 
rem; nec, cum in itinere sine schedis meis plagulas corrigerem^ 
noua moliri ausus sum. quare hoc loco quaedam retractanda. 



*) Addendum, fol. 97 fragmentum (deff., propp. 1 — 4) Cat- 
optricorum exstare (^x r&v %aTonTQL%&v Eif^iXsiSov). 



PROLEGOMENA. XIX 

nam cum antea etiam reliqnis codicibus, inter quos nonnulU 
satis antiqui sunt, aliquid auctoritatis tribuerem, nunc mihi 
persnasi, Uat. 204 solum recensionis fundamentum esse. hoc 
intellexi reperta emendatione loci, qui est p. 146, 16, ubi xairee 
e codd. deterioribus et Uat. 204 correcto recepi, quamquam 
non placuit (u. p. 147 not.); sed seruata scriptura codicis 
Uat. 204 lenissima mutatione egregia euadit sententia: %al 
ziiv avyrjv s^bd^siav olaav (lineam coniungentem mediam lu- 
cemam et rimam tabellae et lucem siue punctum illuminatum 
alterius tabellae rectam esse). hinc adparet, quanta distantia 
Uat. 204 ceteris praestet;*) itaque cum eo scribendum 
p. 146, 26 iiidatsvaav 
p. 148, 9 noXXa] TtoXXdmg 
p. 148, 17 ana] omittendum 

p. 160, 11 ante 6iiiux inserendum ro; Sh TcaQdXXriXa Ta iBlAy IIN, BJ 
p. 162, 10 fLsttov — 11 H^] omittenda (etiam v) 
p. 162, 13 Tial ovvG)] omittenda (etiam v) 
p. 162, 14 Tc^ iisysd^Ti] omittenda 
p. 164, 22 JKZ] JKH 

p. 166, 20 &v — p. 168, 8 (paivstai] ovtiovv t&v &nb rov B 
Hliiiaxog ycQbg rb FK inijtsSov nQOtSTtmtova&v iintivoiv 
li^soDQotSQa iaTl{v) ij BF iinsQ i) BZ' dfioiatg Sii xofl 
inl r&v l|^?. oifKoijv tb fikv F tov Z iistSGjQ&csQov 
(pavsttaif tb $h Z tov J, tb Sh J tov K. ita enim 
cod. Faris. 2342 (nisi quod 6i%tiva)v (istsaQotdtri ieti 
habet) addito tovto t^/fcsi &XX(Qg &nb &XXov dvtLyQd- 

(pov iv TflS ktSQ(p (liQsi iv (srnisico , et ita habuisse 

V e uestigiis pro certo colligitur. tum in figura 
cum cod. Paris. 2342 permutandae Z et z/; p. 166, 20 
pro Bz/, BZ, BK (ita Paris. 2342) exspectaueris BZ, 
BJ, BK, sed cfr. p. 168, 13, ubi e V recipiendum 
puto Br, BJ. 
p. 168, 14 (ov — 21 6Q&tai] oimovv tansivorcdtr^ t&v &nb tov B 
Hfiiiatog nQbg tb dZ ininsdov nQoancntova&v &Ktiv(ov 
ictlv ij Bd, XG^l &n6ytSQ0v (paivstai tb A* tb d aQtx. 
tansiv&csQOv (paivstai tov P, tb $s F toij Z. ita 
etiam Paris. 2342. 



*) Simul adparet, manum rec. nulKus momenti esse, quippe 
quae aut scripturas codicum deteriorum intrvidsAi ^jo^ ^^ 'scss^ 
interpolet. 



XX PROLEGOMENA. 

p. 170, 12 ro^iiTCQoad-sv] to^(i7tQ06&s 
p. 172, 3 B] omittendum 

p. 180, 22 BK, rKJ] BKJ; cfr. Optica genuina p. 36, 23 
p. 188, 28 ijiJLiTivKXiov] rjiimvXLvdQOv 

p. 196, 25 6 B r] 6 nsQl rrjv BV. in figura F ponendum, ubi 

BJ circulum secare uidetur, quamquam ita ob- 

scuratur, eam ad planum circuli perpendicularem esse. 

p. 226, 9 JEZ] 4)7(6 JEZ (pro AEB scribendum {fnb AEB) 

p. 238, 24 ScTCOxaQOvv] &7tox(X)QsirG) ; scribendum &7Cox(OQSt uel 

fortasse &7tox(*iQsTtcci (cfr. p. 110, 26) 
p. 240, 21 Ticci — 22 (psQSGd-ai] omittenda 
p. 166, 7; 180, 23 omittenda, quae uncis inclusi. 

fortasse etiam p. 224, 3, 4, 5, 7, 10 cum V pro J reponen- 
dum A, ne J bis usurpetur; tum A delendum in figura priore 
p. 223 et p. 224, 3 cum V m. rec. scribendum @NA. magis 
dubii sunt loci p. 190, 12 — 14; 196, 3, quia ibi correctio non 
manu recenti facta est; sed crediderim, hic quoque manum 
primam sequendam esse. 

a V proximus abest cod. 6 et sine dubio, ut in Eutocio 
(u. ApoUon. n p. VI), ex ipso V descriptus est. nam non modo 
p. 146, 16; 148, 9; 160, 11; 166, 20; 168, 14; 188, 28; 196, 26; 
238, 24; 240, 21 solus fere cum V consentit, sed etiam saepe 
eosdem errores habet, uelut p. 148, 15; 152, 1; 170, 8, 9; 182, 13; 
192, 24; 202, 15; 212, 4 (u. p. 208, 11), et quae meliora habet 
(p. 148, 21; 160, 9, 10, 14, 19 j 162, 8; 170, 12; 172, 3), prompta 
erant librario illi perito audacique (u. Apollon. II p. VEE); 
p. 148, 17 sine necessitate &y,a addidit, p. 208, 11 falso ABFJ 
scripsit. commemorandum etiam, cod. 6 solum scholia codicis V 
nr. 82, 84, 85 habere et rationem numerorum in 82, de qua 
u. p. 278, e V optime explicari. sed iam supra p. XIX uidimus, 
librarium codicis 6 etiam alium codicem habuisse; cfr. quod 
p. 182, 13 — 15 scripturam manus recentis codicis V in mg. habet 
addito iv &XX(p. etiam exemplaria recensionis genuinae ei 
praesto fuisse, adparet e scholio 21 p. 259, 8. cod. 6 manu 
recentiore correctus est ad similitudinem codicis V correcti 
(p. 144, 1 6 E()%Xsid7ig, ^ yi^vonsvces, p. 146, 12 nv%tLOv, 20 &^iot, 
p. 148, 3 ovv cbff, p. 150, 23 nQbg &vtCX7i'\piv t&v 6Qoct&v, p. 162, 3 
ij KM, p. 172, 9 t&v uQu famv [isysd^&v, omnia m. 2; p. 190, 12 
$L& — 14 6Qad"iJ6Stai mg. m. 2). 

ex V descripti sunt etiam codd. 5 et 10; nam interpolationes 
in eo manu rec. adscriptas (etiam quae in cod. 6 non exstant) 



PROLEGOMENA. XXI 

in textu habent, uelut p. 144, 1, 14; 146, 20; 148, 8; 160, 23; 
288, 1, 20, 22; nec cod. 10 e cod. 5 antiquiore descriptus est; 
u. p. 144, 4 SiTCOQQi.nxoviievag] cod. 5, ytvoiiivceg cod. 10 et V 
mg. m. rec. (y^.); p. 146, 12 tctvxlov] cod. 10, tcvht^ov cod. 6 
et V m. rec; p. 160, 14 iitLTi^dsLov] cod. 6, iniTriBiov cod. 10 
et V; p. 162, 1 rd] cod. 5, roihro cod. 10 et V. cod. 6 autem 
correctus est; cfr. p. 148, 23 ^doftaro?] V, cod. 10, (fdb/iaro? r^s 
P%l6vriq cod. 5; p. 160, 13 fi^v ya^] V, cod. 10, yccQ ^ cod. 6. 
codd. pv e V originem ducere, ostendunt errores com- 
munes*), quales sunt p. 196, 6; 222, 26; 224, 26; 230, 17 et 
p. 214, 11, ubi -4 in V ita formatum est (/f), ut litterae A 
simillimum fiat; cfr. praeterea commune compendium ^p. 194, 19. 
e p. 212, 3, ubi error codicis V et emendatio iuxta se in textu 
posita sunt, adparet, inter V et rp unum saltim arcbetypum 
communem intercedere, et boc ea re confirmatur, quod p et 
cum V contra V consentit (p. 162, 5; 166, 21; 166, 16,17; 168,10; 
172, 7; 186, 6; 194, 9; 198, 2, 10; 218, 11**); 240, 21 praeter 
locos supra adlatos; cfr. quod p. 182, 14 scriptura interpolata 
codipis V in p supra scripta est) et cum V contra v (p. 186, 11 ; 
192, 8; 198, 3; 200, 25; 218, 18; 220, 8; 222, 16; 224, 6; 
230, 10; 234, 22; 238, 11, 14; 244, 8, 16; cfr. quod p. 174, 16 
error codicis p ex ea scribendi ratione codicis V ortus est, 
quam seruauit v). nec in p quidquam reperimus melioris me- 
moriae; nam ra p. 144, 10 leue est nec prorsus certum, p. 178, 
10***) uera scriptura corrigendo restituta est, p. 158, 7 — 8 
errore in p omittuntur, ut e p. 6, 6 adparet, reliquas emen- 
dationes bonas ut p. 148, 16; 232, 13 cum v communes habet. 
p magnopere interpolatus est (p. 166, 20; 190, 12; 212, 24 
?16, 21; 224, 10; 242, 18; corrigendo demum p. 168, 14, 21 
210, 9, 20), interdum e recensione genuina (p. 162, 10, 13 
216, 4); cum V correcto consentit p. 182, 9; 200, 19. etiam v 
et in minutiis quibusdam (p. 208, 20; 216, 8, 10) et in erroribus 
(p. 162, 8; 164,14; 180,9; 198,11; 210,13; cfr. p. 166, 19; 
206, 18; 220, 11; 228, 24; 236, 14), quam arte cum V cohaereat, 
ostendit (cfr. in Catoptricis p. 292, 6; 302, 22; 310, 6; 314, 6). 
et quae emendatiora habet, pleraque tam futilia sunt, ut 



•) P. 182, 2 pro Tdg scribendum at, p. 230, 15 r j KN rj BF 
pro i} KN Ty BF cum Sauilio contra codices. 
•♦) Cfr. p. 86, 3. 
*••) Ibi in adparatu scribendum: dri "V^v. 



YXn PROLEGOBfENA. 

librario tribui possint (p. 148, 3; 150, 9, 14, 19; 162, 4, 27; 162, 2 
164, 6, 11,16, 21, 22; 166,10; 170,8,9; 178,14; 182,13,17,22 
184, 20; 190, 6; 192, 24; 200, 3, 6; 202, 15; 206, 9; 208, 7, 10 
210, 3, 18, 23; 212, 26; 214, 15; 218, 6; 220, 12; 224, 3 SN 
226, 21; 228, 11; 230, 15; 236, 8; 238, 23; 240, 2, 6; 242 
12, 18; 244, 3, 23; 246, 4; in Catoptricis p. 286, 6; 294, 11 
308, 4; 314, 17, 22; locos, ubi V m. 1 uel 2 correctus est, ut 
par erat, omisi); paullo maiora, nec tamen ita, ut captum 
librarii excedant, p. 148, 15, 21; 152, 1; 198, 23; 200, 3, 5; 
232, 13 ; 298, 5. nec aliter exspectandum erat, quoniam etiam 
in Eutocio eadem est ratio codicum Vv (v ibi est w), u. 
Apollon. n p. V. interpolationes p. 162, 14; 166, 7, 16; 168, 10; 
180, 23 codicum vp communes e recensione antiqua petitae sunt. 

e V praeterea pendere uidetur cod. 26; nam p. 194, 19 
compendium _^ habet, p. 174, 16 @N, p. 148, 3 i^SQtfiiisvov, 
p. 172, 7 recte &6ts %af. archetypum communem communes 
produnt cum vp interpolationes et coniecturae falsae p. 146, 16; 
148, 9, 17; 188, 28, item p. 166, 20 sq. et p. 168, 14 sq., quae 
e recensione antiqua petita sunt, sicut etiam p. 162, 14. cum v 
conspirat p. 182, 14 dQmnsvov^ p. 184, 9 rbv ZA, p. 186, 11 
Griiistov, contra v cum Vp p. 152, 20 TCSQicpSQSLCi. origiueni 
interpolationis in v ostendit p. 192, 8, ubi ag r}iiLKv%Xov supra 
scripta sunt, sed p. 146, 12 nvnTCov^ p. 164, 2 Kal TCccQoiXXriXa^ 
p. 168, 14 3)v — 16 d^sSQTina interpolationes in textu habet, 
quas ceteri aut omittunt aut in mg. relinquunt; p. 190, 12 — 14 
in mg. habet ut V, sed manu 1 {rl rb tjiilgv lin. 14 omisit 
lacuna relicta); p. 176, 16 interpolationem e p. 28, 24 petitam, 
quam V recenti demum manu habet, in mg. habet a manu 1 {yQ.\ 
quocum conferri potest scholium 54 e p. 58, 16 — 18 petitum. 

e cod. 26 pendent sine dubio communi intercedente 
archetyp6 codd. 27, 28, 32, e cod. 32 descriptus est cod. 30, 
ut his locis comparatis constat: p. 144, 5 ^vQidtovl 27, 28; 
d^riQlov 32, d^riQtcDv 30 supra scr. d m. 2; p. 144, 11 &noQQi'3crsiv'\ 
28, 30, 32; ciTtOQQCnrsi 27; p. 146, 5 ai] 27, 28; om. 30, 82; 
p. 146, 23 ZXa'] om. 26, 27, 28, 30, 32 ; p. 146, 25 TCQoasnaQ-riGav 
28; p. 148, 7 kcoQ&ro'] k\aiQ&ro 26, bQ&rai 27, 28, 30, 32; p. 148, 14 
oi)8s] iLTids 26, 27, 28, 30, 32; p. 148, 17 afia Sia] ZXa a/ia 26, 

27, 28, 30, 32; p. 150, 7 ra\ om. 28; p. 150, 8 &'no%Xriao%aQ-ai 28; 

p. 150, 10 %arsa%svayi,svaL\ TiarscKSvaay^j 26, narsatisvaaiisva 27, 

28, 30, 32; p. 150, 11 6(pQri6LV 28; p. 150, 13 (pcDvrj] cprivfj 27; 



PROLEGOMENA. XXIII 

p. 150, 20 dcvdQOSidstg 28; i(insvSLv] 27, 30, 32; I^hisvslv 26, 28; 
p. 156, 2 Uov] om. 26, 32; p. 160, d jjtol oi) leaQokXrika supra 
scr. 26, .rixot xal nctQdU,7\ka mg. 27, 30, 32; p. 166, 16 xoQ 
S(i(Latos invjisSfov ^snisvGiv] 26, 27, 28, 32; %si{isv(ov iitvjtsdaiv 
xov diiiiatog 30; p. 166, 20 i)] 6 26, 27; p. 166, 23 ydo] paullo 
obscnrius 26; Ss 27, 32; p. 176, 16 interpolationem codicis V 
in mg. hab. 26, 30, 32; p. 236, 12 rd] x6 26, 27, 32. 

de fragmento Gatoptricorum in cod. 30 hoc tantum notaui, 
p. 286, 1 legi ^ito^sie^^di Bif^Lv e codice interpolato aliquo. 

e codice v descriptus est (in Opticis) cod. 12, ut ex his 

locis adparet: p. 150, 10 &'Kor}v fikv ydo] &'Koiiv \ ydg v, ScKorjv 

yoQ 12; p. 152, 20 ycsQLtpsQSLa] sv^sta y^afifiif v, 12; p. 154, 10 

/* « 

^QoaTtLTttcDGLv ul 67f)SLg V, 7CQ0<S7tL7ttai6Lv al 6if)SLg 12; p. 226, 18 

irei] i\ v, in 12. et Romae scriptus est apud Greorgium Selva 

episcopum tum Francisci I apud Papam legatum. 

codd. 1, 14 a V originem ducere arguuntur loco memoraMii 
p. 314, 1. nam cur ibi pro altero A sine ullo sensu A E habeant, 
causa est, quod in v littera £ figurae prioris p. 311 casu ita 
coUocata est, ut litteram A p. 314, 1 statim sequatur, quasi 
coniungendae sint. cfr. praeterea de Opticis p. 152, 5 a^rco] 
bis V et cod. 1; p. 154, 11 TtQoajtiTttcoGLv] TtQoaTtCeLv v, cod. 1; 
p. 168, 1 Pz/ — 2 to] mg. m. 2 v, mg. m. rec. cod. 1; p. 194, 19 
TtaQaXX^/ikov] -^ v, cod. 1, 14; p. 216, 8 %svtQ(X}] k— uj v, \i\M 1 
(corr. m. 2), e post ras. 14; p. 242, 18 (paCvstaL ^yyLov] om. v, 
codd. 1, 14. nec alter ex altero descriptus esse potest; u. 
p. 162, 16 taa] v, cod. 1, om. 14; p. 174, 27 forca — iatC] v, 
cod. 14, mg. m. rec. cod. 1; p. 190, 12 ^ta — 14 bQaQ"/iastai] 
cod. 14, om. V, cod. 1; p. 216, 5 rtSQLtpsQsCag] v, cod. 14, rtSQL- 
€pSQsUxg %svtQov ^xovtog tb 5(i(ux cod. 1. quoniam autem p. 288, 8 
XQiywva (sic v) in codd. 1, 14 deest, communem archetypum 
inter v et codd. 1, 14 statuere oportet. sed uterque inter- 
polatus est et correctus, u. p. 152, 20 7tSQLq)SQSLa] cod. 1, svQ^sta 
yQafLfii^ V, yQaiifLi^ 14; p. 244, 8 iitl $1 tavtrig] cod. 1, iitsl dij 
ai) V, iTtl d\ a^btfig 14; p. 216, 21 interpolationem codicis p 
habet etiam cod. 14; errores codicis v saepe in cod. 1 non 
inuenimus, uelut p.l64, 8; 170, 7, 13, 18; 172, 17; 174, 4, 7, 11. 

AE illud p. 314, 1 habent etiam codd. 3, 4, 7, 8 (E del.), 
9, 15, 16, 17, 18, qui ea re ex v pendere arguuntur; quo gradu, 
iam uideamus. 

codd. 3, 7 e cod. 1 descriptos esse , os\«T[i^w3DXx \^ Ts^axosis^ 



XXIV PROLEGOMENA. 

loci: p. 150, 15 &^onaXd'6t6av'\ &\7CocXd'£iaav cod. 1, &7tccXd'sUsav S ; 
p. 152, 17 aittb iikv] aiyt6 cod. 1, ai)x6 8, 7; p. 168, 1 rov] 
Tfl5 cod. 1, 7; p. 168, 2 H] fy:a 1, 7; p. 168, 4 z/T] J 1, 7; 
p. 168, 18 ii 8e -— 21 bQaxai'^ postea ins. cod. 1, om. 7; 
p. 168, 21 post bQ&tai add. y.al ScTtoycsQOv (paivstai tb d' tb /J 
&Qa tansiv6tSQ0v (paivstai tov P, tb dh F tov Z cod. 1, 7 

(cfr. V); p. 288, 3 tQiyava] om. 1, 3, 7; p. 294, 11 «•{ 5] © cod. 1, 
^J 3; p. 294, 12 TisvtQOv] K- cod. 1, %vzXov cod. 3, sed corr. 
neque enim cod. 7 e cod. 8 antiquiore descriptus esse potest; 
u. p. 146, 15 t6] cod. 1, 7, tov 3; p. 148, 10 &vay'iiaSoiisvovg] 
cod. 1, 7, &vay%aioiLSvov 3; p. 148, 14 ^(yrt] cod. 1, 7, om. 8. 
cod. 3 ad similitudinem codicis V m. rec. correctus est; u. p. 148, 
20 ad t6 mg. yQ. /tij; p. 150, 23 ad kavtfig mg. yQ. n^bg &vtl- 
Xriifjiv t&v dQat&v; p.l44, 1 ad Stpiv supra scr. m. 1 6 EviiXsidris; 
p. 152, 25 ad %ai mg. m. 1 nsQi; p. 158, 22 post JZ postea 
add. t&v teatv aQa xal tcc l|i}?. 

fortasse etiam cod. 21 e cod. 1 descriptus est; nam p. 288, 3 
tQlyava omisit et p. 288, 17 pro MN solus habet MH, in cod. 1 
autem hoc loco N ita scriptum est, ut litterae H simillimum sit. 

p. 190, 14 uerba dXov tov in cod. 14 paene absumpserunt 
uermes. iam quoniam oXov omittit cod. 8 et lacuna relicta 
cod. 9, ZXov tov lacuna relicta (in qua avtov m. 2) cod. 13, 
e cod. 14 descripti sunt; cfr. p. 236, 8 TCQOGi^vtoDv fisv] icXriaifov 
corr. in itXriaiov 14, TtXriaiav 13 {^Qoai6vt(ov [isv m. 2), nXri- 
ciov 8, 9. eodem pertinere cod. 20 e scripturis uariantibus 
infra adlatis concludi potest; u. p. 144, 14 tolg'] totg (pojtotgy 
del. Torff (p(o, 14, totg post lacunam 13, totg 9, totg (pcototg 8, 20; 

p. 146, 15 s^bQrjaoiLSv] 8, sijQrjacDpLSv 14, 9, 13, svQrjao^v 20; 

p.l46, 18 6vtog] 6vt(og 14, 9, 13, 20, 6vt&g 8; p. 148, 20 t6] om. 
14, 8, 9, 13, 20; p. 150, 4 nai] 14, 13, 20, om. 9, m. 2 cod. 8; 
p. 160, 11 TiatsaTLS^datisv] 8, 13, 20, corr. ex zatsatisvaasv 14, 9; 
p. 150, 18 6a(pQriatv] 8, 6a(pQiaiv 14, 9, 13, 20; p. 152, 1 rLatd] 
9,20, iMctaa%svaaiLSvai 14, nataaiisvaafLSvai 13 et8 (corr. m. 2); 

p. 152, 7 'b^riXotSQa] 9, corr. ex 'bijyriXaitiQa 14, 20, 'btffriXcitiQa 
8, 13; p. 152, 8 Tistad^ai] 14, 8, 9, 20, nsCastai 13; p. 152, 10 
ysyQaiLyi^svrig] 8, 9, -ils- a uermibus absumptum 14, lac. 13 (corr. 
m. 2), ykQ S(iiisvrig 20; p. 152, 28 ^ nsQi] ^v rtSQi 14, 8 (corr. 
m. 2), 9, 18, 20; p. 172, 6 Bd\ 14, 8, 13, z/ 9; p. 190, 14 ^ iatt 
14, 8, 9, 13 (corr. m. 2). 



n^ 



PROLEGOMENA. XXV 

e cod. 13 descriptus est cod. 2 ; nam cum in uniuersum cum 
eius archetypo consentiat, uelut p. 152, 20 ygainiri, ab eo dis- 
cedit, ubi cod. 13 correctus est; u. p. 216, 4 tisvtqov ^x^vtog 
t6 6iiiuc 2, om. 14, mg. 13; p. 242, 18 interpolationem codicis p 
in mg. hafcet cod. 14 m. rec, m. 2 cod. 13, in textu 2 cum 
iisdem erroribus (kavtrjg oi6y,sva t& 6'firfi^Tt); cfr. praeterea 
p. 190, 16 nqoai^%9-(Q'\ iCQorix^Gi 13, 2; p. 288, 3 ri%^(a<sccv\ rj^d^o- 
6CCV 13, 2; p. 314, 1 ^] e corr. 13, A 2. 

e cod. 20 descriptus est, ut uidetur, cod. 29 mendis leuiori- 
bus correctis; u. p. 144, 2 8i6ti\ 14, Zti 20, 29; p. 144, 14 tolg\ 
rotg cpcototg 20, 29; p. 144, 19 t&v GODfLdtcDV ai\ 14, al t&v 
CcDiidtav 20, 29; p. 160, 10 e^d^sta^ 14, svd^sta 20, svd^stav 29; 
p. 160, 18 6a(pQriaLv\ 29, dctpQLCiv 14, 20; p. 162, 10 ysyQaiir- 
li>ivrig\ 14, yaQ si^isvrig 20, yccQ imisvovarig 29; p. 152, 23 ri 
nsQi\ 29, ^i; nsQi 14, 20; etiam p. 162, 21 %SL(isvriv\ nsmsv) 14, 
ftfV) 20, non dubito, quin {lsvslv habeat cod. 29; nam post 
did de suo inseruit to. 

e cod. 9 descriptus est cod. 19; nam p. 148, 16 in '9'fa^a- 
Tor littera s in cod. 9 ita formata est, ut litterae q similis 
fiat; unde ^Qaiidtav cod. 19; cfr. praeterea p. 144, 17 a^^SLv\ 
^SLv post lac. 9, post ras. 19; p. 148, 24 siSd)Xo}v\ 9, stdooXov 19 
p. 160, 2 §sX6vriv\ 9, psXmvriv 19; p. 150, 4 %ai\ om. 9, 19 
p. 162, 16 tov ts\ tovto 9, 19; p. 286, 21 rLoiXa)v\ kvX(ov 9, 19 
p. 288, 21 naQatsd-svtog^ TCaQati&svtog 9, 19. cod. 19 fol. 196 
haec habet: tituli horum Euclidis librorum sunt apud episco- 
pum Comarium nec non in bibliotheca sancti lohannis et Pauli 
Uenetiis, fol. 381 tituli librorum sequentium sunt in libro anti- 
quo . . . in nostra bibliotheca, que est apud fratrem meum 
D. Laurentium. inde sine dubio descriptus est cod. 15, sicut 
omnes fere codd. Philippsiani Uenetiis oriundi sunt. habet enim 
p. 144, 17 a^- in lacuna m. rec, p. 148, 16 ^QaiidtcDV.*) quae 
in margine m. rec. adscripta sunt, ex editione Penae petita 
sunt, uelut post prop. 24: ^^d^sdiQrnia h«'. acpatQa in 9iaatrj- 
liatog 6Q0)iiivri. nviiXog cpaivstai. Mato} yccQ iv acpaiQa reliqua 
ex impresso codice adde. habes ad finem pagine 16". est 
p. 619, 8 ed. Gregorii = p. 16 extr. ed. Penae. ex adnotationis 
forma adparet, codicem typothetae paratum esse; cfr. ad 
prop. 2 : „* 3tL $h 8lsq%6ilsvov, mg. nota a%6XLov, quod ad hoc 
theorema 2™ in impresso habetur codice minime praetermitten- 

•) Cfr. p. 240, 16 ^aa)\ HacQV codd. ^, Vt>. 



XXVI PROLEGOMENA. 

diun. ergo asierisci loco commentarium priorem sequatur*'; 
est p. 603, 3 ed. Gregorii « p. 6 extr. ed. Peuae. ad prop. 7: 
„addidit dominus Dasypodius aliam huius demonstrationem, 
quae ut superiora scholia addatur". ab eo fortasse pendet 
cod. 25; nam in fine habet, sicut cod. 15: to( 7cq6 xm> E^xXe^- 
Sov 6nxi%&v (m nQb x&v Ei^yiXsLdov dntLH&v teXos cod. 9); ad« 
scripsit lo. Pastricius: hic erratur, nam prolegomenorum finis 
est, ubi suppositiones incipiunt. 

cod. 4 e cod. 8 descriptus esse arguitur scriptura TcaQS&rjzoi. 

p. 288, 14, quae inde orta est, quod in cod. 8 legitur TtccQsd^no, 
et eadem scliolia habet. totum codicem 8, quem Petro nepoti 
scribendum dederat, deinde correxit Angelus Uergetius, in qua 
re satis libere egit, uelut p. 160, 24, ubi desunt sl Ss iisrsa' 
q6tsqov, pro %soixo scripsit Xsyoixo et in mg. addidit: Xslitsi' 
si 8s {LTi iv tm a{)ta inmsdoo*^ p. 146, 9 et 10 imtoiii/iv in iv- 
TOfwfv, p. 194, 19 compendium -^ {naQocXX^/jXov) in tov, p. 238, 
13—14 %atcc tiiv F (K om. cod. 8) &si ia tm K cf^fi^rt l6o~ 
tax&s mutauit. interdum cod. 29 usurpasse uidetur, uelut cum 
p. 144, 2 dtdti in Srt, p. 150, 10 s^^sta in svd^stav, p. 152, 21 
TtsmsvTiv in iisvsLv mutat.*) haec omnia quoniam cod. 11 
partim in textu partim (p. 150, 10; 162, 21) in mg. praebet, 
adparet, eum nihil esse nisi exemplar purius et emendatius 
codicis 8 ab ipso Uergetio confectum; cfr. quod cod. 11 lit- 
teram initialem habet, ubicunque Uergetius in cod. 8 para- 
graphum [ adposuit. u. praeterea p. 162, 12 $ta tic ai)tcc 8t\ 
xal ri AN cod. 8, d?} del., mg. yQ. ZXri, AN corr. ia AZ; 8t,oc 
ta avra xal SXr] ij AZ cod. 11; p. 286, 1 vnotislad^G) bipiv slvat 
sijd^stav 8, supra scr. /J — a; ^'iptv slvai sijQ^slav 'bnonslad-a} 11; 
p. 154, 3 Uergetius interposuit 'bnod-scstg, quod recepit cod. 11; 
p. 314, 1 A habet, quia in cod. 8 E a Uergetio deletum est. 

etiam cod. 23 e cod. 8 descriptus esse uidetur; nam p. 144, 
17 uterque ^stv habet pro a^^stv postea correctum. 

cod. 22 e familia codicis 26 uidetur esse; nam p. 146, 23 
3Xa omittit; sed p. 144, 1 habet: dtptv 6 MytXsldrig, p. 146, 23 
tota^nriv om., p. 148, 20 th iir] rj, p. 154, 3 ByvyiXst9ov dnttKol 
^QOt, p. 242, 18 (paivstat. [ist^ov &Qa tpaivstat tb FJ tov FB. **) 

*) Cfr. p. 232, 13 ^SihQTjiia v icvtiatQocpov ro-D itQb ai}tov 
mg.29, addidit Uergetius; p. 240, 11 itstiovcav yoyvt&v] cod. 8, 
ILSi^ovog ytovCag 29, mg. Uergetius. 

**) Haec etiam in p, quod errore in adparatu omisi. 



PROLEGOMENA. XXVn 

rcc d\ (tsliova (patv6y,sva tov 6y,y,atog itQoaidvtog cc^f^dvsad^cci 
9o%oiJ6L. xttl toc ocb^avdiisvcc &qcc t&v fisys&av dd^st •jCQoedysaQ^cti 
tm Siifiati. ^yytov %tX. 

cod. 16 e cod. 8 pendere, ostendunt scripturae ort p. 144, 2, 
ts p. 144, 10; p. 160, 24 sl ds iiststoQOtSQOv om.; sed obstat, 
quod p. 144, 14 totg habet, non tolg (pmolg. cfr. praeterea 
p. 148, 21 TiLvslad^ca, p. 162, 20 yQcc(iiirj, ut cod. 14 alii; p. 242, 18 
habet: xal tcc ilsl^ovcc §avt&v oq&ilsvcc t& binLccxL TCQoadysa^ccL 
9o%ovaL' xal m ccij^ccvoiisvcc ccqcc t&v (Lsysd^&v dS^SL TCQoadysa^uL 

t& 8fLIUCtL. 

cod. 17 e cod. 26 pendet; nam p. 160, 9 ijtoL o^ TcaQdXXriXu 
supra scr., p. 176, 16 dxQLg — t& B mg. habet ut cod. 26, et 
praeterea eadem scholia praebet (nr. 38 et 41 in textu) et in 
nr. 21 easdem scripturas (p. 269, 8 — 11, p. 260, 14 — 18). iam 
cum p. 254, 17 sHtcol habeat, non sHnjjy ueri simile est, eum e 
cod. 28 descriptum esse (e codd. 27, 30, 32 pendere non potest 
propter ov p. 160, 9); p. 162, 20 i^ svd^sta yQccniirj habet ut v, 
sed in mg. iv dXXtp ri icsQLcpsQSLcc. 

cod. 18 sine dubio e cod. 14 pendet; nam p. 144, 14 habet 
tolg fpoitolg {tolg (poi- del.) et eadem scholia continet; p. 242, 18 
habet: %ccl tcc (Lsiiovcc havtolg old^LSva to5 6iifiatL iTcav^dvs- 
a^aL 8o%ovaL' xai tcc av^avo^isva aQa t&v ^sys^&v 86^sl tcqoc- 
dysad^ac ttp 6iLiuctL' ^yyLov xtX., ut cod. 14 (iavti}g). de cod. 24 
nihil notaui, nisi quod p. 286, 1 habet 'bTcoytsia&o) bipLv. Edi- 
tionem*) Opticorum „cum notis mss." in bibliotheca Uniuersi- 
tatis Paris. adseruatam (Omont, Inventaire m p. 355 nr. 66) 
non uidi. 

Bestat, ut de scholiis pauca addamus. 

praeter codices in adparatu usurpatos, quorum deteriores, 
qui obiter tantum inspecti sunt, fortasse praeter notata unum 
et alterum scholium etiam ceterorum habent, minora prae- 
sertim, in his codicibus scholia insunt: 

cod. 14 nr. 13, 16, 7, 18, 19, 36, 33 + 34, 38 (a p. 266, 10), 
41, 50, 64, 65, 56, 60, 67, 58, 63, 67, 71, 75, 73, 76, 80, 81, 
86, 91, 92. 

cod. 18 eadem habet eodem ordine praeter 7, 36, 60, 73, 80, 81. 

cod. 20 nr. 10, 13, 7, 15, 18, 19, 33 + 34, 38 a p. 266, 10, 
41, 54 + 66, 66, 67, 73; cfr. cod. 14. 



•) Sine dubio Penae; ea enim 48 pagin.aEliafe^\.^\asi.^"^ 
optricis). 



XXVm PROLEGOMENA. 

cod. 4 nr. 7, 10, 13, 15, 18, 19, 33, 34, 38, 41, 54, 55, 56, 
60, 57, 58, 63, 67, 71, 75, 76, 86, 91, 92; cfr. cod. 14 et 8 (« r). 

cod. 17 fol. 107—109' (Opt. 7—24) nr. 19, 21 (ut R) ad 
p. 260, 10, tum nr. 23, tum partem reliquam nr. 21, 26, 34, 37 
et in textu nr. 38 totum, 41; fol. 248-251' nr. 1, 2, 6, 3, 10, 
13, 9, 7, 11, 8; fol. 252 — 253, ubi repetitur prooemium cum 
iisdem scholiis et in textu nr. 5, 4 et propp. 1 — 6, nr. 10, 13, 
7, 16, 18. nr. 38 cum R consentit. 

scholia codicis 1 (Vat., Vat. m. 2, non Vat.*) sola sine 
textu habet cod. Paris. suppl. Gr. 12 chartac. s. XVI fol. 36—40' 
(Omont m p. 202—203) ex ipso cod. 1 descripta; u. p. 267, 15 
tdg (alt.)] comp. 1, tf^g suppl. 12 postea correctum; p. 276, 10 
IksQSotg] lac. 1, suppl. 12; p. 282, 12 mats'] ats post lac, 
13 ro5 dG] t ad^, p. 283, 1 t-n^tav] sHTtat suppl. 12, omnia ut 
Vat. ; cfr. Om Scholieme til Euklids Elementer p. 34. eandem 
coUectionem scholiorum habent Ambr. J 84 inf. chartac. s. XVI 
(ex officina Uergetii; u. Om Scholieme til Eukl. Elem. p. 34) 
et Magliab. XI, 11 chartac. s. XVI (u. Vitelli p. 550), cuius 
pars media eadem omnia continet, quae suppl. 12. 

Homm scholiomm pars antiqua, quam V a manu 1 praebet 
(10, 15, 20, 23, 27, 29, 30, 34, 36, 39, 40, 45, 46, 48, 62, 53, 
55, 56, 67, 58, 60, 63, 66, 66, 67, 68, 69, 71, 72, 76, 80, 82, 
83, 84, 85, 86, 87, 88, 89, 90, 91, 92), sine dubio orta est e 
studiis Byzantinomm, ut cetera scholia operum in Parao astro- 
nomo comprehensomm; saltim saeculo X antiquiora sunt, ut 
ex erroribus codicis V adparet (p. 255, 22 ; 284, 1 ex compendiis 
ortis, p. 270, 15; 272, 5; 280, 15; 281, 17 in litteris; cfr. p. 271, 14; 
280, 11; 281, 17; 282, 3, 4); p. 275, 22 ex Herone citantur, 
quae nunc non legimus. sed cum errores haud ita multi sint, 
crediderim, ea non nimis multo ante V scriptum confecta esse, 
fortasse saec. IX, quo studia mathematica reuixisse docui 
Bibliothec. Mathemat. 1887 p. 34 sq. accessemnt saec. XTTT 
codicum 14 et 26 communia nr. 7, 13, 18, 19, 38, 41, 60, 54, 
75, 81, quae habet etiam cod. 1 exceptis nr. 7 et 76, sed 
praeter 50 (falsum), 64 (« Opt. ant. p. 58, 15), 81 (= Opt. 
ant. p. 82, 12)*) a manu 2, et 33, 73 (cod. 14), saec. XTV nr. 42, 



*) Etiam nr. 75 ex Opticis genuinis p. 84, 22 petitum est. 
conferri potest, quod in cod. 13 manus 2 adscripsit p. 114. 20 
- 115, 9 et p. 116, 22—118, 5. schol. nr. 7 est Opt. ant. schol. 4, 
nr. 33 cum Opt. ant. schol. 31 congmit. 



PROLEGOMENA. XXIX 

4:4, 47 (citationes), 74 (= Opt. ant. p. 84, 5), quae praebent 
codd. 1 et 26 (excepto nr. 42), nr. 21, 26, 37, 61, 62, 64 (cod. 26), 
nr. 8, 17, 22, 24, 28 (= Opt. ant. 10), 49, 79 in cod. 6, qui 
etiam pleraque reliquorum habet, fortasse etiam nr. 31, 32, 51, 
69, 70, 77, 78 (V*). reliqua recentissima sunt (v^ V* saec. 
XV— XVI). 

m. 

De fatis Optioorum. 

Optica, qualia hic e codice Uindobonensi maxime primo 
loco repetiuimus, Euclidis esse, non est, cur dubitemus (cfr. 
Weissenbom Philolog. XLV p. 54 sq.). sed cum recentiores 
tantum exstent codices, mirum non est, locos nonnullos tam 
corruptos esse, ut uerba Euclidis restitui nequeant; u. p. 2, 
1—2; 18,14—16; 34,17; 58,10—12; 84,18—20; 116, 17sq.; 
etiam p. 88, 6 'post itcca&v deest: t&v yaivi&v t&v ^SQiexoiisv(ov 
{jnd, ut legitur p. 66, 23; 68, 18; sed hic error Theone anti- 
quior est, quoniam is non modo p. 220, 15 idem habet, sed 
etiam errorem propagauit p. 204, 15; 206, 2. etiam p. 120, 6 
aliquid turbatum est et fortasse nimis audacter ti^g dh &nb 
t&v 6niidt€ov e Theone p. 246, 1 recepi. cfr. praeterea in locis 
subditiuis p. 98, 23; 114, 6. nec desunt interpolationes ; neque 
enim dubitari potest, quin demonstrationes alterae ab Euclide 
profectae non sint (u. V p. LXXEX); pleraeque e Theone inter- 
polatae esse possunt (p. 36, 4 = Theon prop. 22, p. 48, 9 = Theon 
prop. 28, p. 92, 20 = Theon prop. 43, p. 98, 6 = Theon prop. 45), 
quamquam hoc quoque fieri potest, ut Theon iam utramque 
demonstrationem habuerit alteramque elegerit; non habet &lX(ag 
p. 34, 20; 112, 23; 114, 10. subditiuum praeterea scholium 
p. 50, 1—8. et ueri simile est, etiam p. 64, 4 — 21 e Theone 
p. 202, 1 — 16 interpolata esse; nam idem aliter demonstratur 
p. 76, 12 sq. (omisit Theon), nec in prop. 34 locum habet, ubi 
de omnibus diametris aequalibus agitur (p. 60, 15); adcedit, 
quod uerba fiifra Hcag yfovlag nsgvixovaa p. 64, 26—26 (= Theon 
p. 202, 20—21) propter p. 64, 4—21 necessaria minus recte 
adduntur, quia semper cum binis diametris aequales anguli 
efficiuntur, nec apud Pappum VI, 80 leguntur. itaque puto, 
non modo p. 64, 4—21, sed etiam iirjts Haag yoDviag TCSQiixovaa 
p. 64, 25 — 26 e Theone inteipolata esse, et deindA d^^^i^^o^^s&^ 
intellecta ngbg i^g Ttoisl &vi(Sovg ycaflag p. ^4^*^.1 ^ot\.\iaXi^ 



XXX PROLEGOMENA. 

Pappus et Theon); ita demmn ordine ac ratione progreditur 
demonstratio. 

in codice Uindobonensi Optica genuina cum Elementorum 
libris I — Xy et Phaenomenorum recensione antiqua coniuncta 
sunt, sed quo tempore hoc corpus compositum sit, incertissi- 
mum est; nam in cod. Laur. XXVULL, 3 haec pars tota sae- 
culo XY suppleta est, et in Bodleiano B ne Optica quidem 
tota conseruata sunt. fieri potest, ut in Uindobonensi demum 
haec opera sint coniuncta. 

a Pappo Optica inter opera ad xbv &atQovoiJioviisvov rd- 
srov*) (p. 474, 3) pertinentia in libro VI tractantur titulo non 
addito. VI, 80 — 97 propositiones 34 — 36 cum lemmatis**) 
additamentisque (VI, 87, 88, 92, 98—103) suis retractat (VI, 
90 — 91 = prop. 34, 93— 97 =» prop. 35) et in uniuersum eandem 
demonstrationem habuit ac nostri codices. 

recensio recentior, cui praemissa est praelectio, quo iure 
ad Theonem referatur, u. Studien iiber Euklid p. 139. primus 
hoc suspicatus est Angelus Uergetius, qui in cod. Paris. 2468 
adscripsit: tb nCQOoitiiov i% t^s rot) @scov6g icxiv i^riyrjasms, et 
est coniectura satis probabilis, quamquam Theon in avvra^, 
Ptolemaei p. 7 ed. Basil. prop. 4 ita citat, ut ad recensionem 
antiquam propius adcedat (xat p. 6, 11 habet, om. recensio 
Theonis p. 158, 13; p. 6, 12 Sucan/jiiatog , sed &7Coat7jit,octog re- 
censio Theonis p. 158, 14); ex ceteris, quas citat propositiones, 
nihil concludi potest (prop. 3 in avvt. p. 7, prop. 5 ib. p. 8, 
prop. 23 ib. p. 266, prop. 26 ib. p. 199). de recensendi ratione 
Theonis u. Studien p. 146. intactas reliquit uel leuiter mu- 
tauit propp. 1, 2, 3, 9, 83, 34 (= 34 -f 35 Theon), 41 (= 39), 45 
(=46), 47, 62 (=61), 53 (=62), 64 (=63) et definitiones 
praeter primam; magis mutatae nec in litteris figurae solum 
propp. 20, 37 (= 41), 38 (= 42), 43 (= 44), 60 (= 49), 51 (= 60), 
66 (=56), 57 (=56), 58 (=57); prorsus mutatae propp. 29, 
30, 31 eodem mutationis genere et 65 (= 54). in mutando 
semper fere breuitati studuit (propp. 4, 6, 6, 7, 10, 11, 32, 44 
, p. 98 = 45); demonstrationes ita decurtatas saepissime a par- 

*) Titulum 6 ^LTLQbg &atQovoiioviLSvog scholiasta demum 
Pappi habet. 

**) VI, 80 usurpatur p. 68, 16, VI, 81 p. 68, 6, VI, 85 
p. 74, 15 sq., VI, 86 p. 80, 1 sq., VI, 89 p. 74, 20; 78, 18. VI, 82 
— 84 idem demonstratur, quod in lemmate p. 66, 18 — 70, 17. 



PKOLEGOMENA. XXXI 

ticula oifKOvv incipit, uelut propp. 12, 13, 14, 15, 18, 19, 21, 
26, 26, p. 200, 22 «= p. 62, 19 sq.; cfr. p. 214, 16 = p. 80, 11; i 
p. 174, 6, 17; 180, 26; 182, 23; 188, 6; 242, 16. etiam prae- 
parationem saepe breuiorem reddituelut p. 160, 26 (=p.l0,8sq.), 
p. 184, 18 sq. (=«p. 42, 1 sq.), p. 186, 7 sq. (= p. 42, 24 sq.), 
p. 188, 1 (« p. 46, 2), alibi. prorsus omisit non modo p. 32, 24 

— 86,4; 46,14—48, 8; 60,1—8; 90,18 — 92,19; 96,14—98,6; 
112,24 — 114,18, de quibus locis u. supra, sed etiam p. 68, 21 

— 70, 17; 74, 23 — 76, 16, propp. 46, 49. multo rarius aliquid 
addidit uelut p. 162, 7 (= p. 10, 24), p. 164, 9 (= p. 14, 7), 
p. 196, 2 (=a p. 66, 19; hic rursus oifnovv illud suum usurpat); 
prop. 40 inutilibus ineptisque ambagibus dilatauit. e Pappo 
VI, 80—81 interpolauit p. 206, 21—208, 10; 206, 5-20, fortasse 
etiam p. 202, 21 — 22 ^siSav ds ^ iXdaaav tfig i% xov KSvtQOv 
e p. 668, 16. p. 210, 20 i} iiiv — 23 ro5 O interpolata esse 
arguuntur non modo ipsa forma molesta, sed etiam leiomate 
Pappi VI, 86. propp. 37 — 41 infelicissime sic ordinauit: 41, 
42, 38, 40, 39. 

cum recensio Theonis in 79 fiL%Qa 6LaxQovo\LovnBvcp tradere- 
tur, cuius causa fortasse et ea et recensio noua Phaenomenorum 
facta erat*), Optica genuina non prorsus ab usu mathemati- 
corum remota sunt. uelut Georgius Pachymeres saec. XTTT ea 
in quadriuium suum recepit teste Paulo Tannery Rapport sur 
une mission en Italie p. 39 (Archives des Missions 3® s^rie XIII). 
ex hac parte geometriae Pachymeris et opticis Heliodori 
Larissaei Angelum Uergetium non sine &aude composuisse 
Damiani Optica, quae edidit Erasmus Bartholin Paris. 1667 
e cod. Barber. I 131, demonstrauit idem Tannery 1. c. p. 40. 
itaque quae illa editione confisus de ratione, quae inter Damia- 
num et Optica Euclidis intercedat, exposui Studien ilber Euklid 
p. 187 sq., nunc de Pachymere ualent, quem codicem nostris 
similem habuisse, mirum non est. Heliodoro uel Damiano 
relinquuntur I, 1 — 13 sola genuina. ibi cap. 6 citatur Opt. 
prop. 1: TCQhs xb tov axoixBCov xov Xiyovxog' oi^dev x&v 6^£o- 
fiivtov &ILU oXov bQ&xm. cap. 1 idem argumentum adfertur, 
quod in praefatione Theonis p. 160, 9 sq. ; etiam cap. 8 cum ^ 
Theone p. 146, 24 sq. comparari potest. uterque sine dubio 



*) Mutationes temerarias operum eo pertinentium sigToifL- 
care uidetur Pappus VI, 1. 



XXXIV PROLEGOMENA. 

cum radiis applicatis. uterque angulorum cadit in sexoicircu- 
lum. quare linee applicate ad circumferentiam, quia faciuni 
angulos rectos cum lineis ductis a centro, erunt contingentes. 
quare protracte non secabunt circulum. si*) igitur perueniat 
radius longior, erit, quod due linee recte includant superficiem; 
quod est impossibile. quare relinquitur, quod linee longiorea 
sunt contingentes. 

post p. 77, 14 (u. not. crit): demonstratum est in AP libro 
Euclidis**) elementorum geometrie circa***) datum trigonum 
circulum describere. quare possibile est uolenti circa pxl\) 
trigonum et adhuc circa Tce}) sectionem. descriptis autem tribua 
sectionibus manifestum, quoniam duarum maior pml sectio,. 
at uero xl-\\) minor quidem eafff), maioruero olx. propterea 
uero maior qui m mxl sectione angulus; in minori enim*f) 
portione angulus maior; qui uero ad x maior quam qui ad n. 
scilicet ab eo quod est demonstratum usque huc non est d& 
libro isto, sed extra sumptum [est enim Opt. ant. schol. 70]. 

alia quoque huius generis scholia in mg. habet D, sine 
dubio e codice Graeco petita; sed magis memorabilia alia 
uidentur scholia, in quibus alius interpretationis mentio fit, 
cuius uestigia etiam alibi deprehendimus ; nam in L similia in 
textu post protasim leguntur praemissis uerbis habet alia trans- 
latio (in D fere praemittitur alia translatio; ibi in mg. sunt 
m. 1) et eadem fere in cod. Oxon. Colleg. Corp. Chr. 283 (in 
textu, in alia translatione habetur). **f ) aliquanto plura uestigia 

apparet semicircumferentia. si enim bzg esset 
semicirculus, cum db et dg sint linee con- 
tingentes circulum, utraque facit angulum 
rectum cum bg diametro per XV 11 tertii Eu- 
clidis. ergo triangulus bdg duos rectos habebit 
angulos; quod est impossibile. 

*) Scholium: si enim dicamus cadere intra, 
esset hoc contra caudam pauonis, si autem 
extra, erit, quod due et cetera. **) Hic e 
textu interponuntur p. 77, 14 (g)it — apparebit. 

***) Sic. cod. Torun., contra D. f) kxl 
Torun., qui omnino in litteris discrepat. 

ff) kxl Torun. fff) ed D, eo Torun. *f) ffic del. 

sectione D. **f) ffic codex binis columnis scriptus est, in 

priore propositiones , in altera demonstrationes , quae breuis- 
simae sunt nec cum genuinis quidquam commune habent. in 
fine fol. 164^ legitur: nota, quod sexaginta et tria toreumata 




PROLEGOMENA. 



XXXV 



praebet cod. Bibliothecae Gymnasii Torunensis R W^ 2 (scr. 
a. 1359, u. Curtze Zeitschr. f. Math. u. Phys. 1868, litt. Abth. 
p. 45 sq.), qui recensionem continet a genuina multo diuersiorem 
et interpolatam. ex eo codice hic subiungam, quae etiam in 
DLC (0 = cod. Oxon. Coll. Corp. Chr. 283) inueni, adiectis 5 
horum scripturis uariantibus (p. 3, 13 — 14 habet T: omnes uisus 
equeueloces esse, qui secundum equales angulos deferuntur, 
non autem sunt equeueloces, qui secundum inequales lineas 
deferuntur. non sub quocunque angulo rem uideri. inde a 
prop. 28 magis ad D adcedit et praeter interpolationes mi- lo 
nores eandem recensionem praebet). 

prop. 1: Nullum uisorum simul totum uidetur. in eodem 
instanti non uideri plura. esto uisum ad, oculus uero b. dico 
igitur, quod non simul comprehendetur a uisu ad secundum 
se totum. incidunt radii ba, bc^bd; bt uero sit perpendicu- 15 

laris super ad. quoniam igitur in 
triangulo bcd angulus bcd est 
rectus, erit per 17 primi maximus 
angulus illius trianguli; quareperl9 
eiusdem ei opponetur maximum 20 
latus. recta igitur linea bd longior 
erit recta linea bc. et eadem ra- 
tione ba longior bc. resecetur ergo 
per 3. Euclidis ad equalitatem bd 
quidem in puncto e, ba uero in 25 
puncto f. quoniam igitur omnes 
uisus transpositi secundum equales 
lineas sunt equeueloces, in equali- 
bus partibus defertur uisus ab oculo b ad tria puncta ecf. 
uisus quidem delatus a b super lineam bd citius fertur ad e so 

continentur in isto libro. Aimare, gratias age, quia hoc opus 
sic glosulasti sub magistro lohanne de Beaumont. explicit 
feliciter liber de uisu. 

12. aKa translatio. nullum uisorum simul totum uideri D. 
habet alia translatio. in eodem instanti non uideri plura LC 
ceteris omissis. 14. igitur] ergo D. 16. incidunt] incidant 
enim D. 16. igitur] ergo D. 18. primi Euclidis D. 21. 
igitur] erg& D. 22. eritl est D. 23. longior est D. bc] 
bc eadem D. 24. per 3. jEuclidis] om.D. bd] ipsius bd D. 

26. igitur] ergo D. 27. transportati D. ^^. ^^^^^iSwy^ 
temporibus D. 30. quidem] autem D. 




XXXVI 



PBOLEOOMENA^ 



quam ad d. eadem. ratione ostendetur, quod citiug uidetur . . 

per antepenultimam;, que est: omnes uisuB equeueloces esse, 

qui secufltdum equale» liiieafi deferuntur, non autem sunt 

est. hic igitur similiter delatus snper lineam ba citins per- 
5 ueniet ad f quam ad a, qnare in tempore breuiori trans- 

portabitur uisus ad punctum c quam duo pnncta a et d. eadem 

ratione ostendetur, quod citius uidetur c quam qoodlibet 

punctum in linea dck patet igitur, quod puncta linee dOj 

quaato propinqui^ra simt puncto c, 
10 tanto citius a uisu comprehendentur; 

unde punctum g citius quam punc" 

tum d et punctum h citius quam 

pimctum a. protractis enim lineis 

bg bh, cum ai^pilus bgd sit ex- 
15 trinsecus ad angulum bcg, eritmaior 

ipso per 32. piimi Euclidis. anguius 

uero bcg est rectus; quare angulus 

bgd erit obtusus, quare erit maxi- 

mus angulus in triangulo bgd per 
20 17. primi. quare ei opponetur maximum latus per 18. primi. 

linea igitur bd maior erit linea bg. quare per predictum mo- 

dum demonstrandi citius fertur uisus ad g quam ad d et 

similiter ad h quam ad a. oum igitur punctum c citius com- 

prehenditur a uisu quolibet puncto linee ad et ei citius uici- 
25 niora quam remotiora, successiue igitur comprehendetur linea 

ad Q> uisu. non igitur simul, quod foit demonstrandum. 

Notandum igitur, quod de rectis lineis et de superficiebus 

planis intelligenda est propositib, de lineis autem curuis et 

superficiebus concauis sperarum non est hoc necessarium^ quod 
80 proponitur per 1. propositionem. si enim in centro circuli 




1. eadem — 4. igitur] om. D. 1. uidetur] segi comp. 

incertum T. 3. qui] corr. ex que T. sunt] seg. compp. 

dubia T. 4. delatusj om. D. 6. breuiore tempore D. 6. 
duo] ad duo D. eadem] eadem etiam D: 7. uidebitur D. 

8. aliud punctum D. igitur] iterum D. 9. puncta D. 

11. unde] ut D. 15. ad] om. D. bcg\ bgc D. 20. 
17.] 19. D. 18.] 17. D. 21. igitur] ergo D. 23. igitur] 
ergo D. comprehendatur D. 24. uicimora citius D. 26. 
igitur] ergo D. 26. igiturl ergo D. totum simul D. 27. 
igiturj est autem D. et de] et D. 30. 1.] istam D. 



PR0LE60MENA. XXXVU 

xdsnB ponmtur, «ius periferia ciHnB simul comfNrehe&detiir, cum 
omnes linee, per quas dirigetor nisus, sunt eqnaies, et similiter, 
si in centro spere poneretur oculus, tota eius concauitas simul 
in eodem tempore irisui apparet. 

prop. 2: Equaliiim ma^tiMlinum in distantia iac^tium 5 
pr(^us posita pergpicacius nidetur. 

eqnailiiiin nisibii^m inequaliter in eandem partem iacen- 
tium tiel remotomm propinquiori est uisus certior. 

sint nisa ad ee, que oportert ymaginari 
«sse equalia eit paralellogramma, oculus uero lo 
sit hd^ ad uero «it propinquHis oculo quam ce. 
dico, quod ad peri^icacius uidetnr quam ce. 
incidant enim radii hd ha. hc he, positis 
notis fg^ ubi he hc intersecant ad. quoniam 
igitnr ad uidetur sub angulo ahd, quare ^ 
uidebitur sub tribus angulis ahf fhg ghd, 
snb qoorum uno uidelicet sub angulo hcg 
uel bgf comprehenditur cg. sub pluribus igitur angulis uide- 
tur ad quam ce. per 13. igitur petitionem huius perspicacius 
et certius xddetur ad quam ec. et hoc est propositum. 20 

prop. S: Unumquodque uisorum habet longitudinem spatii, 
qno £aeto iam non uidebitnr. 

quodlibert tiisibile per elongationem aliquam non posse 
terminare uisum. 

sit res uisa ad^ oculus uero h, radii uero prouenientes ad 35 
terminos rei uise sint hahd. quoniam igitur in ultima petitione 




1. uisus] om. D. ponatur D. eius — simul] oculus simul D. 

2. sunf| essent sibi inuicem Z). 8. ponatur D. 7 — 8 DL, 

7. uisibilium TL, magnitudinum D. iacentium uel] om. 
DL. 8. remotarum D. propinquioris DL. 9—20 D; L 
habet p. 6, 8 — 12, sed pro p. 6, 12 trigoni — 16 aliam demon- 
strationem, qiuze in mg. tramit. 11. hd] h D. ad] da D. 

12. dico ergo D. IS. hc he] he hc D. 14. secant D. 

15. igiturl ergo D. 17. hcg] ehc D. 18. hgf] ghf D. 

ۤ} ce D. igitur] ergo D. 19. 13. igitur] 6. ergo D. 

hiiiu«] D, i T. 20. uidebitur D. et hoc] quod D. 

23-— §4 D, cuiuslibet uisibilis per elongationem terminari 
tiisum L. 2S. aliquam] om. D. 25 sq. D; L hahet p. 5, 19 
— 7, 6, «cd pro p. 7, 2 ad — 4 uidebitur interpolationiem, 

25. peruaiientes D. 26. igit\iT"\ eigo D. X3&aaa\'^ Ti. 



XXXVm PROLEGOMENA. 

positiim est, rem sub quolibet angulo non uideri, erit accipere 
aliquem angulum, sub quo semper non uidetur res. 

sit igitur angulus abd minimus angulus determinatus uisui. 
elongetur igitur ad magis ab oculo et ec, que equedistet ad 
5 in priori situ, et ducatur be bc. quia ergo angulus ebc minor 
est angulo abd, angulus ebc non erit determinatus uisui. 
quare non incident uisus ad ec. quare non uidebitur ec, cum 
positum sit in 4. petitione, ea uideri, ad que uisus incidit, et 
ea non uideri, ad que uisus non uadit. da igitur habet longi- 
10 tudinem spatii, quo facto iam non uidebitur; quod est pro- 
positum, et demonstrabimus per illa qua 2. 

quae sequuntur, in T in textu sunt post protasim, quae 
fere cum D consentit: 

prop. 4: Equalium uisibilium super unam lineam eodem 
15 puncto coniunctorum, quod remotius est,minus apparere (ow.Di). 

prop. 5: Inequalium. quod [propius, uidebitur maius, cum 
uersus eandem remoueatur partem inequaliter (L). 

prop. 6: (Equedistantium linearum magis remotum minus 
apparet interstitium (L). 
prop. 7 : Equalium spatiorum super eandem basim existen- 
tium, quod propinquius est, maius reputatur (om. DL). 

prop. 9: Quadrata per distantiam apparent rotunda (DL). 

prop. 10: Partes inferiores plani remotiores uidentur 
altiores (D). 
25 prop. 11: Superiorum plani superiores partes secundum 
uisum declinare (DD).' 

prop. 12: Per recessum, que dextra sunt, sinistra uidentur, 
que uero sinistra sunt, dextra uisualiter adire per totum (DL). 

prop. 13: Equalium equalis altitudinis sub oculo iacentium 
30 remotius uidetur altius (Di, <m. T). 



1. rem] rem non D. quolibet] quocunque D. non] 
om. D. 2. quo semper] minori quod D. uidebitur D. 

3. igitur] ergo D. abd] adb D. angulus] om. D. 4. 
igitur] ergo D. et ec — 11] om. D extr. pag. 11. illa] 
seq. comp. dub. T. 16. equalium L. propiusl propinquius 
est L. 17. partem remoueantur L. 18. equioistantium L. 

20. spatiorum] despatorum T 22. apparei^] uidentur DL. 

25. superiorum] superioris DL. partes superiores L. 27. 
per recessum] precessum D. sunt] om. D. uidentur] om. L, 

28. uero] om. L. sunt] om. D. per totum] partem DL. 



PROLEGOMENA. XXXIX 

prop. 14: Super ocnlum consistentium quantitatum et 
«iusdem magnitudinis, cuius maior est remotio, eius maior 
putatur dimissio (JDL). 

prop. 15: Cum super idem planum similiter steterunt in- 
equalia, quod radio capud minoris contingenti punctoque sub- 5 
teriori de maiore concluditur, minus cum lumen inclinatum. 
sensus utriusque translationis est, quod propositis duabus 
quantitatibus inequalibus ut ab et gd, et ab sit maior et gd 
sit minor, quod, quanto oculus magis accedit ad ^c^ minorem, 
tanto ab uidetur minus excedere gd, et quanto magis recedere lo 
A minori, tanto maior magis uidetur excedere minorem (2/, 
om. JDT). 

prop. 16 : Quod in directo . uerticis ipsius ultra minorem 
4e maiore positione altius est eo, quod oculo uidetur tiltiori, 
inequalibus in uno stantibus plano (DL, pm. T). 15 

prop. 17: Si uieus unius altitudinis remanserit, exdistantia 
non mutatur proportio (2/, om. DT). 

prop. 18: Altitudinis quantitatem per umbram solis et 
rectam uirgam similiter inuenire (DL). 

prop. 19: Erecta uirgula speculoque interposito, quanta 20 
sit altitudo paralella, dicere (L). 

prop. 20: Qualiter profunditatis certitudo sit habenda (i, 
T in mg. praemissis uerbis alia translatio). 

prop. 22 : Si fuerit oculus in eodem plano cum arcu, circum- 
ferentiam uideri rectam (X, ut alia translatio mg. T). 85 

prop. 23: Quod de spera cemitur, eius medio minus est 
et uelud circulus (mg. T, praemissis uerbis habet alia trans- 
latio, ut solety L, in quo sequitm: propositum est, quod minor 
pars medietate spere uidetur ab uno oculo). 

prop. 24: Alia. quanto magis accedit, minus de spera 30 
cemitur, et id maius apparet (i, mg. T). 

prop. 25: Alia. uisiones, quarum distantia dyametro spere 
par et equidistans fuerit, regunt oppositas secundum dyametrum 
notas (2/, mg. T). 



2. eiusdem] unius D. ^i^s] om. DL. 3. reputatur L. 

demissio DL. 19. rectam] r, erectam similiter L, om. D. 

similiter inuenire] T, repperire L, cognoscere D. 20. 

uirga L. 26. quod] comp. e corr. T. eius medio] cuius 

medio T?, medio eius L. 31. apparet maius L. 32. dia- 
metro L. 33. equidistans] i, equidem T? xfe^xjcoS^WikjgQsi^la. 



XL PfiOLEGOICEKA. 

prop. 26: Si maius diametro fuerit interstitiiUDL lasioimm^ 
uidebitar medio spere maius (C, om. T). 

prop. 28: Medietate minus aspici de oolumpna (€, om. T% 

prop. 29 : Quod a propinquiore de oolumpna rotunda minua 
essentialHer oemitur, maius est apparenter (LC, om. T). 

prop. 30: i^amidis medietas rotunde non uidetar «b 
oculo super ebadum basis collocato (Z, gm. T). 

aliam rursus recensionem oontinere uidetur ood. AmbroB. 
P 21 sup. saec. XIV; ine. radius egreditur ab ooulo ^uper line»» 
equales/ des. fol. 188 : ex loco uisus ad o^tram circuli seoiui-- 
tdum dispositionem, quam dixdmus. et hoc est, quod demon*- 
strare uoluimus. explicit liber de aspectibus Euclidis feliciter. 
titu^lus e£rt;: 'tiber de aspectibus et specuUs Euclidis, cum in 
x^eteris codicibus fere inscribatur de uisu. prop. 9: figuve 
ortogonie, cum aspi^iuntur a longe, uidentur rotundae. 

Quae UiteUio in libro YV cum Eudide conunnnia habet, 
neque in propositionibus nec multo minus in d^nonstrationibua 
ad uerbum oum uUa ha^m infterpretationum consentiunt. 
Bogerus uero Baco exemplaria nouit, ubi utraque int^rpretatio 
coniuncta erat; u. Op. maius p. 246 (*== Perspectiua ed. €om- 
bach p. 115): in libro de uisu hoc idem uuH auotor, cum dieit 
in X propositione : rectangulae magnitudines e distautia uisae 
peripheriae apparent [p. 17, 6]. sed quia reetaAgulae figurae 
huiusmodi non possunt esae nisi aequilaterae, ideo alia trans- 
latio subiungit: quadrata per distantiam apparent rotunda 
[u. p. XXXVm]. cfr. ib. p. 246 (« p. 116 Combach): auctor 
iibri de uisu et multi aestimabant, magnatudineim comprehendi 
per quantitatem anguli apud oculum, unde in principio illin& 
libri supponitur, quod uisa sub maiori angulo af^arent maiora 
et sub minori minora, et sub aequalibus angulis nisa apparere 
aequalia [p. 3, 6sq.]. lohannes Pecikham (Perspectiuae com- 
munis libri tres. Colon. 1592) altera interpretatione urtitur; u. 
I, 89 non sub quocunque angulo rem uideri [p. 8, 14] ; cfr. I, ^8^ 
de hac certitudine loquitur Euclides de uisu, ouminquit: nullum 
uisibile simul totum uideri, sed per iromutationem pyramidis. 

Per totum igitur medium aeuum sola Optica genuina in 
manibus hominum erant; nam quae XJincentius Bellouacensis 
Spec. nat. XXV, 45 habet praefationi Theonis consimilia*), e 



*) et ex hoc concludit Euclides, quod uidemus per lineas 
ab oculo egredientes, uidelicet per triangulum, cuius basis est 



PBOLBGOMENA. XLI 

^ITemesio (Nemesii Yersio Latina ed. Holzinger p. "80) habere 
pote&rt, nt snadet exemplnm nnmmi iis commnne (apnd Theonem 
•est .feTidvrj) , qnamquam hic Eudlidem non nominart, sed „geo- 
metros". renascenrtibns uero litteris recensio Theonis per- 
un%ata est. 

£a nsus est Geoigius Ualla, ^qui Be expet. et fug. rebus XV, 3 
pAortem «Opticorum Latme uertit (u. Neue Jaihrb., Suppl. XU 
p. &94 — 395); nam non modo praefartioinem Theonis haJt>et, sed 
^eitiam demonstrationes xecehsionis Theoninae, nelut iprop. 10: 
posittis in£ca Qcnlnm planis, qnae remotiora snnt, sublimiofra 
uideaitnr. sit nanqne oculus b supra ck planum coUocatum, 
a quo saae oonlo cadant radii hc bd hf hk, qnorum ^k per- 
pendicnlaris sit in coUocatnm planum. aio, cd ipso df snb- 
limius nideri. igitnr cd ipso df sublimiora nidentnr, sA fd 
quam fk. ergo qnae sub subliniioribns radiis ajq^arent, sub- 
limiora compatrebunt. ex hoc loco simnl adparet, cuius generis 
Qodex eius fuerit; omittit enim p. 166, -22 t6 6i — p. 168, 6 
>^ Jl Z nt cod. Monac. S61 et apographum eius ood. Paris. 2852; 
-etiain p. 168, 8 fpocuvdfjbsvec habuzt pro .6^ntva cuan Monac. 861 
^ p. 288, 19 SuKpiQTfCfu (differat) pro &uxtpalvritcci, cmm Paris. 
^852 (et sine dnbio etiam Monac. 861). iam oum Monac. 861, 
<ut moz uidebimus, Uenetiis aliqnadido fuerit, ubi Ualla degebat, 
neri simile est, eum hnnc ipsum codioem habuisse; et sdioHa, 
qnae Ualla recepit (nr. 7, 10*), 15**), 18, 88 a p. 266^ 10, 41, 
60, 5J7, 58, 91, 92), in Monac. sunt, nr. 6iO, 57, 58 eodem ordine, 
cum Paris. 2352 scholia non habeat.***) 

XnlteEpretationem integram primns edidit Uenetiis a. 1505 
Bartibolomaeus Zambertus, qni de codicibus snis haec dicit isL 
praejGEutione: cnius quidem disciplinae rationem quandoque cum 



res uisa, et angulus expansns est m oculo, eiueqne diameter 
auper partes rei in se discumt, ne apprehendamns partem 
uisibilis, ndsi quam diameter attingit, ideoque dicit, quod non 
atatim ^uidemus denaaium in panimento iacentem, quod etiam 
probatur per demonstrationem. 

*) Hoc sine dubio etiam in Monac. exstat; habet enim 
cod. I^id. 

^*) Iflieipit: aliud sit itaque; m Monac. est: ^ijlo V- 
***) fiehqna additam^ta Uallae, quae commemioraiai 1. c. 
p. 394—395 (post prop. 10, ad propp. 19— ?11\ «caa 5«2isv^ «^^ 
Marte aliunde snmpsit. 



XLH PROLEGOMENA. 

apud Socraticum Euclidem in uetustissimis et tineis ac carie 
contritis Graecis codicibus legerem, quodam stupore perfusus 
hominis ingenium arduum et sublime inde diiudicans opus 
illud mira solertia sed maximo studio non legi, sed relegi 
transcripsique pariter, ut tanta doctrina quoque inter nostros 
codices summa ueneratione seruata reperiri posset. iam cum 
cod. Leid. manu Zamberti e cod. Monac. descriptus sit, eum 
sine dubio*) hic significat; et cod. Monac. re uera „tineis et 
carie" pessime habitus est; quem tum Uenetiis fuisse, hinc 
iure colligimus. cod. Leid. igitur ei in interpretando ad manus 
fnit, et concordant scripturae, uelut p. 144, 14 lucentibus 
illustrantibusque ignibus, p. 148, 20 sub uisum namque cadit 
spectatae rei imago, p. 152, 1 fidemque huiusmodi efficiunt 
in praesentia radii, p. 152, 20 quae linea est; inquit enim, quod 
eo quia in uisu linea manet, p. 190, 14 oXov om., minus est 
et, p. 236, 8 propinquum. cfr. supra p. XXTV sq. 

Editio princeps prodiit Parisiis a. 1557, 4° per loannem 
Penam, qui de codicibus suis haec dicit: itaque cum mihi 
essent aliquot exemplaria Graece scripta, quae Petrus Eamus 
Philosophiae et Eloquentiae Regius professor atque idem alum- 
nus tuus [Caroli Lotharingi Cardinalis] et praeceptor meus ab 
amicis mutuo acceperat, nolui Eempublicam diutius hac Euclidea 
doctrina carere. fundamentum editionis est cod. Paris. 2350; 
nam pleraeque coniecturae Uergetii a Pena receptae sunt, uelut 
p. 144, 2 ort, p. 146, 9 et 10 ivTOfiifjv, p. 146, 21 pr. %al om., 
p. 148, 26 icnBQQSi (&7tSQQ£Sv rl &^sqqsi Uerget.), p. 152, 21 
tisvsiv; non recepit sv&stav p. 150, 10, unde constat, eum 
codice Paris. 2468 usum non esse; cfr. praeterea p. 146, 11 
tovrm] Pena, om. 2468, p. 148, 17 dicc tb nLvsiad-ai] Pena, om. 
2468. scholia 13, 15, 18, 19, 91, quae omnia in Paris. 2360 
insunt, in textu habet; sed praeterea multo plura interpolauit, 
quae in nullo codice inueniuntur; apud Gregorium sunt p. 606, 
3—17; 607, 1—8; 608, 16—26; 611, 40—45; 612, 13—22, 37—47; 
618, 17-33; 617, 15—30 {i% t&v tov ndTCTtov)**); 618, 22—25 
et alius generis p. 619, 8 — 22; 626, 25 — 36; 627, 32 — 34; 



*) Obstare uidentur temporum rationes, si recte compu- 
tauit Weissenbomius (cfr. V p. CIV), quod quo modo expli- 
candum sit, nunc non diiudico; satis nuhi est, cod. Leidensem 
ante interpretationem editam finitum esse. 

**) Titulus fictus; apud Pappum nihil eiusmodi. 



PROLEGOMENA. XLm 

-628, 1^8 et conatus alicuius mathematici Graece docti (an 
Rami?)*) uidentur esse. 

Praefationem propositionesque solas deinde Argentorati 
«didit Cimr. Dasypodius a. 1571. editione Penae usus est; 
nam p. 164, 23 habet: tavta [ihv oiv 'bnoytsiGd^oD rjiitv, ^| &v 
^oc l|^ff d^scDQi/jiiccta dsix^V^stav, quae est interpolatio Penae. 

Etiam Gregorius editione Penae nititur. inspexit hic illic 
^codicem Bodleianum nescio quem, e quo nihil fere protulit, et 
Sauilianum, nisi hunc e notis Sauilii tantum citat; inter codd. 
Sauilianos Bodleianos nullum repperi; p. 623 not. 2 e Eeg[io?] 
xtdferuntur p. 194, 19 — 20, quae omisit Pena extrema pagina 
et cum eo Dasypodius et Gregorius. cod. Sauilianus codici 
XJatic. 202 similis fait; cfr. p. 168, 14 atv — 16 &S(i)Qr}iia] in 
textu Sauil. et 202, p. 242, 19 ante Isyyvov add. ybsliov &qa (pai- 
rfstai tb T^ toH FB, tcc fisi^ova §avt&v oiSiisva tov 6inMxtog 
3CQOGi6vtog inav^dvsG^at doTiovai. %al ta a{)^av6iisva &Qa t&v 
jLsysd^oiv do^SL nQoadysG^ai rco ^iniati' Sauil., 202; in fine 
uterque tiXog s^Xritps ra jCQb t&v 6nti%&v E^yiXsidov, p. 156, 12 
interpolationem codicis V (m. rec.) habuit, p. 164, 10 propriam 
(xal tfig KA iXaGaav). habuit scholia 10 (p. 606 n. 1), 37 
(p. 617 n. 2), 41 (p. 618 n. 2) praeter 7, 75, 86, quae Gregorius 
in textum recepit; ea omnia in Uatic. 202 exstant. 

Schneiderus denique (Eclogae phys. I p. 381 — 391) codices 
non habuit, sed Gregorium sequitur paucis additis coniecturis. 

IV. 
De Catoptricis. 

Etiam in Catoptricis unicum fundamentum editionis est V; 
inde enim pendet v, ut supra p. XXI exposui, ex v rursus Mm 
pendere arguuntur loco illo p. 314, 1, de quo dixi p. XXIII; m 
enim AE habet, M uero JE errore latius manante. lamen 
«08 abiicere nolui, ut manifesto documento pateret, quo modo 
interpolatio in his opusculis studiose lectitatis paullatim in- 
cresceret. M non ex ipso v, sed e cod. Uat. 192 descriptus 
«st, quoniam p. 288, 8 tQiyoava cum eo omittit; cfr. p. 292, 21 
BZJ] V, mut. in BZ Z^ m. rec. v, BJZ 192, M; p. 296, 22 
iaai — 23 ya^viai] V, om. 192, M; p. 302, 26 t&v d^tpsoiv] Vv, 



*) Cfr. de Petro Montaureo, alio eius discipulo, ApollorL. XL 

p. xvn. 



XLiy FfiOLBQOMfiKA. 

am. l^, M. interpolatio modice ^grasaata est^ uelut p. 286, 8; 
288, 5; 290, 21; 292, 1, 4; 298, 5; 302, 26; 804, 4, 15; 310, 4; 316^ 
15; 332, 10; 340, 23; ofr. p. 328, 10; p. 2^0, 13 &v recte addidit. 

iam ex iUo AE ueri simile est^ m quoque e Uat. 192 de- 
scriptum eese, et hoc coufirma/tux «rroiibuB quibusdam cum IT 
commuziibus., uelot p. 300., ±2; 306, 28; 308, 17, 1«; 312, 16; 
830, 18; 838, l^ qui ad commnmem archet^pum retfereudi sunt;. 
itaque rglY^ova p. 288, 8 coniectura addidit, aicut t&v '&if)B&it 
p. 302, 25 alio loco. nam m ab homime haud indocto sermo- 
niflqiue tmathemaitici «atis perito per totum opus audaci&Bime 
interpolatuB est, ut adpacatus oriticus quauis pagina docet. 
addendae hae sGnpturae errore in adparata omissae : p. 288, 1 
post B add. nai m, 4 post i^xdtoaav supra scr. yo^, 5 FJT] 
T^ riJL, AK'] tijv AK, ^ vniiuiTo] (>M6%eirai^ post ii^ supca. 
«cr. iazl, 7 &Qa'] &Qa iarlv^ 9 firrco ^ postea oorr. in iLllk 
^ &Rra), 15 Hati — 1« £] xai. ^tdsI £ari iai/iv ^ im^ MKB^ 
16 Z] i)nb NKJ^ rj A] 4mh VMK r^ ijnh AKN, 17 E — ZJ 
vnb BKr r^ imb dKA, 18 lirij iath, 20 E] imh BKT, 21 Z] 
irxh JKA^ 22 0, E] Mt BKM. harum interpolationum ple- 
rasque (desunt p. 266, 19; 294, ^; 296, 4; 298, 22; 800, 8, ei 
quae in m postea additae snnt p. ^88, 4, 6, 9) in Y adscripsit 
manus recens (cum eiTore. FMK pro FKM p. 288, 16) nouis 
de suo additis p. 296, 5; 300, 4 (partem tantum mutuata est 
p. 300, 4, 15); sed inde a p. 300, 18 taedio laboris inutilis Ab 
incepto codici praestantissimo pulcherrimoque funesto destitit. 

de codicibus, qui etiam Optica continent, in cap. 11 dictum 
est. addendum, cod. Ambr. A 101 sup., Paris. Gr. suppl. 186,. 
Uindob. suppl. 9 interpolationes codicis m habere p. 288, 4, 6, 
6, 7, 15, 16, 17, 18, 20, 21, 22, 23; itaque in Gatoptricis inde 
pendent. cod. Uindob., de quo infra uidebimus, praeterea 
problema de dnabqis mediis prc^ortionalibus aliasque notas 
mathematicas cum eo communes habet (u. Apollon. 11 p. XIX); 
p. 288, 16 FMK habet postea correctum. Paris. suppl. 18^ 
p. 314, 1 AE habet, p. 288, 16 FKM, p. 288, 6 &Qa iati {&ga 
Uindob.). cod. uero Ambros. locum dubitandi dat; habet enim 
p. 288, 6 &Qa iarl, 9 &Ua dii ItfTo» cum m correcto, 16 FMKy 
sed p. 288, 8 t^joiva omittit cum Uat. 192 et M; sed ardie- 
typus oeterorum esse nequit; nam p. 288, 1 inlnsSav hontQov 
habet, 20 yavla omisit. 

Catoptrica sola continent hi codices: 

1) cod. Marcianus Gr. 302, chartac. s. XV (Elem. I— XIII^ 



PfiOLEGOaiaiSUL XLT 

DsA». ciim Maidno, Theodosii Spliaeidc&, Phaenomena, Cat- 
<iptncai, Barlaam, Ptolemaeum), magna ex parte a Bessarione 
43criptus. 

2) cod. Paris. Gr. 2013, chartac. s. XVI,. ex parte a Chri- 
stophoro Auer scriptus (Omont 11 p. 179). Catopkic» habet 
foL 81—97. 

3) cod, Paris. Gr. 2448, bombyc. s. XIV (Omont 11 p. 263). 
€atoptiiica habet fol. 50^70. 

4) God. BeroHn. Philipps. Gr. 1543, chartac. s. XVL fol. 
1 — 12^ CatOptrica, fol. 12^ — 14' tres notae mathematicas (ut 
TJindob: snppL 9), quas. ex m edidi Zeitschr. f. Math. u. Phys. 
XXXTIT hist. Abth. p. 161—163 et Eucl. V p. 720 nr. 2. 

5) cod. Berolin. Philipps. Gr. 1544, chartac. s. XVI (Elem. 
I — ^XIII, Data cum Marino, Theodosii Sphaerica, Phaenomena, 
fol. 243^248 Catoptrica). in principio bis: a^r} ii §i§X/os rov 
^sdsgUiov ToH Mrflxcfticxov %al t&v &X7i&&g^ q)iXovvTiav. 

6) cod. Bibliothecae marchionis de Bosanbo 370, ohartac. 
s. XVI (scripsit Angelus Uergetius, continet Catoptrica)- u. 
Omont, Catalogue des mss. gr. des d^partements p. 72. 

7) cod. Archiuji historici Toletani 29, chartac. s. XVI (€at- 
optrica, Elfementa). 

de cod. 6- et 7 nihil aliunde notum. cod. 1 ex m de- 
scriptus est; nam omnes eius interpolationes habet (p: 286, 19; 
288, Isq., etiam 4 ya^, 6 iavl^ 9 &XX& Si} ictm). inde descriptus 
est cod. 5, ut iam ex indice operum, quae continet, satis ad- 
paret. et cod. 1 Malatestae cuidam, sine dubio possessori 
codicis 5, mutuo datum fuisse, ostendit fMorellius Bibl. ms. 
p. 178. easdem interpolationes habet. 

cod. 4 ex ipso m descriptus est; interpolationes habet, 
etiam y^g p. 288, 4, sed neque 6 icvL neque 9 &XXic Sii iaTm. 
prorsus eadem ratio est codicis Uindob. suppl. 9, et cum 
praeterea easdem notas mathematicas ex m petitas contineat, 
ex ood. 4 descriptus est; cfr. etiam p. 288, 9 9rji] 9i Uindob., 
cod. 4. 

cod. 2 et ipse interpolationes illas habet (etiam p. 288, 4 ya(>, 
6 iatl, non uero 9 &XXa di} I(F7<o) et ex ipso m descriptus est; 
p. 314, 1 AE habet, p. 288, 16 FMK. quare e Paris. suppl. 186 
descriptus esse nequit; immo huius archetypum esse cod. 2, 

adparet e p. 288, 23 Xotit'^'] m, Xoi' cod. 2, Xot7c6v Paria. 
suppl. 186. 



XLVI PROLEGOMENA. 

cod. 3 cum 8ui generis sit, hic collationem plenam sub- 
iungam, quamquam ad emendationem nihil fere inde peti 
potest. 

EifTiXsldov TiatwctQMcc. p. 286, 1 a' add. — Ttdvta] om. 
3 j}' add. — a7tavta'\ ndvta 4 y' 5 yivovtai 10 ^' 19 Ix- 
%v^% 20 ^B&QTi^a a 21 iitvjtiitfQV p. 288, 2 ^'] ^i 5 ifniv 
om. TK] triv JA ii JA] ovt(og i^ BK AK] tijv AK 7 rc5 
t6 9 mg. m. 2: iv t& ivdittQip tm xv^tco — MvontQov] om. 

13 A] A yoavta 18 ^ari iativ 19 mg. m. 2: iv t& TioiXtp — 
^rf] om. 20 BK] KB 21 imTciSat ivdntQov tov KM 22 r<yi] 
^i iiiai] G}v tari ^^^'^ 23 Xoim^] «al >lot7rdt' — r|f] Xoijcfj tfj — 
^atai] ietC p. 290, 4^ AV] AKT 5 rcrrjv noiovca ytovCav 6 
rijv] corr. ex tf^ m. 2 10 idBixQ^ri] vTtSTisito — yaovia] yoD extr. 
lin. 12 iativ] om. — kavtfjg 13 d^fidtfs^ av] naC 21 oItt* 

22 olJr' p. 292, 4. BK] BK dcvaTiXaiiivri 6 ini] %al ini — 
naC] xal t&v 9 ^^r^] «ijrd 13 di] S' 17 B^T] BF^ 19 
E, ^] z/, E 20 iv t& xvprdS ^vcJjrr^ca 21 8i] d' p. 294, 1 
insSsvx&fa] ^d-eo 8'6'd'fiAa] om. 2 xal ^x^£^Xiftf^fi]] om. Iwf/} 
xal ^jrf^ 4 ^«yrt ftfi^^cov] iisiSfov iatC 6 JT] om. 7 -4] in ras. 

8 HE] rj I 10 rov %6vtQov 17 ariiiBioig totg A, J, F] AJF 
arinsCotg 18 i^fitxvx^ta 20 aviinsasttai 22* 6'fifuif ii/ r^ Trs^t- 
q>SQsCa p. 296, 1 ^£] ^' 3 insC] %al insC 4 iisiitov] fisiioDV 
%aC 6 ft£^|;ov$] iisi^ovsg 6 ffcf^ in ras. — iXdttav — &Qa 

IL&XXov 7 xara tb ^] om. 8 diioCoag ds 11 fiEffc^ 12 ns~ 
aoHvtai 13 ^f] 8' 17 ^x/Sf/SXijV-O-cotfav] iJx^(oaav 20 r^A:] 
rjS:® 22 fftfiJi;] om. 23 siaiv] om. p. 298, 3 ^ari] ^arj iatC 
6 OPZ diioCoig rc5 ^r^^ ro^ov d'soi}QrjiLatL &no8sC%vt)tat 6 i^] 
^ i^ 9 iXdtt(ov 12 rifv] r^ 13 Ss] om. 20 ^f] d' p. 300, 1 
aXXoag mg ^d&og q^aivstai 2 AV] AA*) 11 dvsatQaynisva] 
-atQ- in ras. 16 ^d^og] pd^og {lsv 22 oikcos p. 302, 2 ovx- 
ovi; 6Lvav,Xa-] in ras. 5 ovt<og 8 dTrdrs^or dnotSQOv 13 z/] 
z/ at B^z/ BFE 14 ds] d' 22 ^' a^rcov p. 304, 1 M] H — 
A]@ 6 ^ffrco ndXtv—^tb KG p. 306, 1 ovrcog 5 BAE BFJ 



*) Ad prop. 9 praeter nostras haec figura 

7 



j^£ 



o 



et 



PROLEGOMENA. ^ XLVH 

7 tfjg] tov 9 E, J] JE 10 A] F — F] A 13 Idstv] i&sa- 
»fjvocL 16 $s] $s ietGi 20 insSs^dx^o)] %^co — AM!S}Z] 
AMNS2 21 rix^maav 22 B^] BT p. 808, 3 tatg] &Qa 
tatg 5 TflS (utxumque)] corr. in r<5 6 &XXd — T] ISI — foij 
ieti 7 N] T — is:] N S BS] BT 10 iativ] &Qa — <Jva- 
TLXaa&rjastai 18 ^e] ^i} 20 TcXsvQccg ^x^v 22 (Sii^ayey^ag)'^'!» 
aw^ tfjg AB p. 310, 1 TCoXvyoivov 5 nsla^oaaav 6 imSsvyw- 
Hsvais 16 ^i xat] ^e p. 312, 2 Z] Z &Qa 5 ovv] ^^a 18 
si)^sta] om. 24 Sfia p. 314, 6 avn§oXrjv &7t6] om. 7 roii] 
ro 14 ^3r«j6v;^^fij] rjx^oi 17 rj ^E] r^s JE p. 316, 4 xal 
iyipspXi^ad-aiaav] om. 7 -4J8[] Jf-4 10 t&v (utrumque)] om. 

12 xat] dri %al 13 rw] om. — 8] © 16 J&] @J 17 0Z 
c^'d'frai/ 21 ^Xattov p. 318, 1 ra] sic 2 yisvtQOv 3 ^x- 
/Sf/?Xi}<y^fl)<fai/] rix^oiaav S VAK] A ^ corr. — ^E] E 9 ^rot- 
ovtftv 11 Jf] P 12 EJ5: r))ff J5:Z] ET tfjg PZ n&g iLsit<ov — 
^&XXov] &Qa iisiStov p. 820, 1 Tivgtov] om. 2 Ttaqa%sia^ai — 
3 ^r] om. 3 antonsvGiv 9 tfjg NI] &Qa tfjg N IQ BKA 

p. 322, 3 iXdttova*) ^ iXdtt(ov 6 tfjg] tfjg nsiiovog 8 ^Xar- 
rovog 10 iXdttovog 11 iTtstsvx^oa] i]x^(ii» 13 tsiivst — yoivi&v 

15 $s] ^ — inl t6] om. 18 t&v BEJ 20 iiarrcoi; 21 
rj] sic — oo/jfcff] om. p. 324, 2 5t/;fcg &va%XanLSvri 15 ^f] ^' 

17 inTtsasttai — &va'KXanisvri ij 19 ^Xarroro^ 26 iXdttovog 

p. 326, 8 %d'] corr. ex xy'**) 9 ts^^j tb oV^a 12 ^ari] tari 
iativ, l- in ras. 14 &Qa ti6vov 18 &^g] tsd'^ 20 ^T 23 
iW6>] @M p. 328, 3 ^r^dff] al itQ^g — iyivovto 4 yiyvr^tai 
— avii§aivovta 5 r6 ^fifia] om. 6 yivsa&ai 8 iii^aXmv 9 
dvaydyr^g] corr. ex &ydy'ijg 13 oiJrf rcov ^i^rog mg. 19 «ii/a- 
xilcoi^rai — cbff] om. 20 rd] seq. ras. 1 litt. p. 830, 9 6i})ig] 
-i- in ras. 12 iTtsSsvx^oiaav] %al iTatsvx^&aiv 18 &Qa] om. 

15 @E] 0E ovtGig 18 ii6vov] om. — Ixarf^ov] -xare- in 

£0/ 

ras. — E] E t&v s F 22 F, A] AT p. 832, 3 &yaya)v] -a- 
e corr. 4 ft£(roi/] corr. ex y,iaaiv 5 dtafttr^ov] TtSQtqiSQsiag 9 
a-^rj] sic 12 6>iff — 13 0] om. 16 JfB, K@] K@ KB 16 
^-O-iDoav iJx&Gi 17 BT] TB 18 iatlv rj P] iatl nal i} E — 
lislSoiv] om. 19 o^x] xal oiyi 22 rj] rofg 23 ^atoiaav] corr. 
ex forco m. 2 p. 334, 10 fi(Jco>loi/] -X- e corr. 11 ^vdsrr^ov] 

*) Ubi nihil adnotatum, uocabulum ^iarrcov plerumque 
compendio scriptum est. 

**) Li numeris propp. ab ls' ad xy' numetvsL^ ^ci^\fcTv«5ft \ss. 
rasura est, item in numeris xe' — xti' et in W ^xc> V ^.^'^^^^^i^- 



XLVm '^ PBOLEOOHENA. 

in ras. 21 (papsltccL — ^i] ^' 22 BT] corr. ex Bzf p. 336, 2 
yo^] apo; — iVf^] BA 3 d^BAgrifioc %&' postea add. 5 nal'] 

om. — iieffSv 10 r] r xal higcc Htpig ij FJ ScvaTiXaiitivri inl 

rb B 11 r&v] corr. ex rco 12 ^^u-d"© 16 iieaSv — nqoa&itov 

xal ro-O iv6yCTQ0v p. 338, 3 ebs] sic 4 iXattov 6 ^siiigrma X"^^ 

mg. 8 TtQoaa corr. in ngoaa}' 9 ra ^' ^Xarrot^o; 10 r&it] r» 
— ra ^f] rco #^ rcc 11 rAr — &Qurt6Qd (alt.)] onii 12 ydg] 
om. — ^M] ^B 13 yivowt' 14 ZJT® 16 AM] KB — 
iV^] iy 17 ^e^rrova 18 Ttavto&avcci p. 340, 1 Xa\ a e corr. 

6 iyifiepXi/jad-Gi] %^co 7 z/T] sic 10 iXecttav — r^^] sic — 
XoKtijg] tfj Xomfi 11 BVJ] BAJ 13 AJ &iitlg] JA 14 rff] 
6m extr. lin. 17 elg] 7teQL(peQ8lccg elg 21 Slcc] di p. 342, 2 
avtai 3 TiivTQOv yccQ 4 srotoi^crt 5 yivovtai 9 '9'€(i|LicM.i'0fi«Va>v 

10 atwemov] -^n- in ras. EvxAa^dov xarosrrpMc&if rale^. 

harum disorepantianim pleraeque interpolationem apertam 
prae se ferunt, uelut p. 294, 2; 296, 1; 304, 13 (cfr. p. 328, 13, 
ubi interpolatio nondum in textum irrepsit); quae proba^ilia 
habet (p. 314, 6 &7t6 om.; 318, 1; 322, 21; 332, 9; 338, 3; 340, 7, 
forfeisse etiam p. 340, 21 di pro ^a), coniecturae tribuenda, 
sicut iam in m nonnulla eodem modb correcta sunt. nam 
arctam cum T necessitudinem ostendunt loci, quales sunt 
p. 298, 5 et p. 335, 15 if ; interdum etiam cum deterioribus 
consentit, ut p. 326, 18; 330, 18. 

Scholiorum longe maxima pars eiusdem aetatis est, cuius 
antiquiora ad Optica, h. e. saeculi IX, ni fallor; errores in V 
(p. 350, 6; 351, 9; 353, 19; 354, 2, 7, 9, 20; 355, 1; 357, 4, 11, 
16, 24; 358, 23; 359, 4, 7, 8, 10, 18, 25; 360, 6, 10, 11, 16), 
quorum nonnulli ex compendiis male intellectis orti sunt 
(p. 351, 4; 354, 2, 6; 356, 11; 360, 6), ostendunt, ea aliunde 
sumpta esse; et duos mJTiimum fontes eorum fuisse, adparet 
ex 51 et 52, quae idem eodem modo demonstrant. saeculo XY 
adcesserunt 2, 9, 11 (V^); nam recepta sunt in p, cuius librarius 
initio scholia codicis V descripsit, sed mox destitit; ipso V eum 
uBum esse, ostendunt signa, quibus scholia ad textum referuntur, 

in 3 ^, in 7 *i); eadem enim habet V; p. 349, 5 compendiosam 
scripturam codicis V male intellexit. de suo addidit nr. 1 et 
praeterea ad St,' kavt f]g p. 290, 17 a>g inl tfjg dQ^^g^ ad iXda- 
aovog p. 290, 18 ijyovv tf^g d^elagy ScXXcc inl tfjg iiel^ovog driXov6ti 



PROLEGOMENA. XLIX 

^on/jfSSL riiv &vdyiXa6iv ^yovv tfjg &ii§Xslcigj ad inl vb B p. 292, 1 
ijyow iq)' kavti/jv. omnia fere scholia codicis V (nonV^) habet 
etiam q (nr. 49 ex eo enotatum non est, sed fortasse iniuria; 
p. 350, 27 fort — 351, 2 omisit), sine dubio a manu 1; ad 
p. 286, 1 habet yiocrcc noivbv tb 'bTtOTisiad^a) , ut V. scholia q^ 
ante cetera scripta simt; nam ab eorum coUocatione locus 
scholiorum q pendet.*) 

Catoptrica genuina non esse, exposui Studien iiber Euklid 
p. 151, nec ante Proclum quisquam ea nominat. de erroribus 
eorum in rebus expositis u. Studien p. 150 et Gregorius fol. c"; 
et forma quoque demonstrationum parum adcurata est. iam 
hoc confirmare licet comparato loquendi genere cum Opticis 
genuinis, quae magnopere differunt. uelut de radio oculi in 
genuinis Opticis usurpatur &%tlg 74®« (+ 3 in locis subditiuis; 
in propp. 18 et 20 est radius solis), HtiJig 20®« (-|- 2 in locis 
subditiuis p. 84, 20; 36, 13. undecies legitur in deff. et propp. 
1—3, ceteri loci sunt p. 16, 27; 54, 4, 17, 23; 58, 3, 4, 7, 8; 
116, 5. oculum significat p. 30, 17; 42, 27; 56, 10). in Opticis 
Theonis**), quae onmino breuiora sunt, proportio mutata est; 
Bijjig enim 20®« usurpatur (de oculo p. 194, 19), &7Ltig uero non 
plus qnam 52®«. Theonem uocabulum o^tpig praetulisse, mani- 
festum est, si comparauerimus p. 20, 8 &%tlvsg et p. 170, 7 
6^sig. in Catoptricis denique uicit btpLg (cfr. definitio p. 286, 1), 
quod 70®« legitur, cum &'M!ig nusquam compareat (nam in 
prop. 30 est radius solis). eadem prorsus ratio est uocabuli 
oi)yLovv in principio demonstrationis. in Opticis genuinis legi- 
tur 15®« fere et p. 36, 13 in loco subditiuo, apud Theonem 50®« 
(cfr. p. 80, 11 i&v &QCC et p. 214, 16 ovnovv Ztav\ in Catoptricis 
dimidio fere breuioribus 22®«. adcedit ratio angulum per unam 
litteram significandi (i^ A, non i] Tcgbg t& A\ quae apud Eucli- 
dem inaudita est (in Opticis genuinis non inuenitur nisi in 
loco subditiuo prop. 42 aUog); apud Theonem in propp. 8, 29, 
42, 43 usurpatur, in Catoptricis uero saepissime (propp. 1, 2, 
3, 4, 5, 6, 13, 14, 15, 21, 24, 25, 28, 30). 

*) Li figura p. 341 in V desunt litterae E, z/, Z; addidi 
e p. 340, 4 — 6. littera P ita in V posita est, ut in figura 
nostra, et ita eam habuit scholiasta p. 362, 2 ; sed debuit intra 
angulum collocari ut II. 

**) Praefationem non respexi; in ea dxr/g 10®« le^tur, ^-j^ig 
quater (p. 148, 12, 18; 152, 27; 154, 2); ibi fere significa»t <^<i7^- 
lum (p. 146, 20; 148, 1, 4, 20; 152, 4, % 10, 1^,"^!% ^^.^A^.'^.,N>>. 
^acJides, edd. Heiherg et Menge. YII. ^ 



L PBOLEGOMENA. 

his perpensis oritur suspicio, Catoptrica, qualia nunc 
habemus, a Theone demum compilata esse, ut cum eius re- 
censione Opticorum in tbv iimq6v &inQovo(Lovii,8vov reciperentur; 
nam credibile est, eum in Opticis, ubi opus genuinum ob 
oculos baberet, a sermone antiquo minus deflexisse, et quae 
in Opticis nouare incepisset, in Catoptricis demum ad finem 
perduxisse. boc si uere suspicatus sum, in cod. Uat. 204 
tbv iiiTiQbv &(rcQovoiio'6fLSvov talem habemus, qualis a Theone 
compositus est; ab initio Catoptrica non comprehendit (Studien 
p. 152). tum causa est dubitandi, scripseritne omnino Cat- 
optrica Euclides; neque enim hoc ex p. 30, 3 concludi potest 
(potest enim etiam alienum opus ita citare), et Proclus in Elenu 
p. 69, 2 fortasse iam Theonis opus in manibus habuit, in quod 
Euclidis nomen ob uicinitatem Opticorum facile transferebatur. 
in opere suo componendo Theon uti poterat et Archimedis Cat- 
optricis, quae habuit (in Ptolem. synt. p. 10; cfr. schol. nr. 7), et 
Heronis. et reuera p. 286, 17 — 19 ex Archimede*) citatur ab 
Olympiodoro in Meteorol. 11 p. 94 ed. Ideler, et prop. 4 ad 
uerbum fere apud Heronem legitur prop. 7 (Eose, Anecdota II 
p. 322 ; cfr. ibid. prop. 9 =« Catoptr. 24, prop. 10 = 5). 

sed cum ceteri Catoptricorum libri Graece non iam exstent, 
nostrum opusculum aetatem tulit, quia in zbv (iMQbv icatQovo- 
ILOvftsvov receptumerat; is enim propter Ptolemaeum semper ab 
hominibus Byzantinis lectitabatur; cfr. Theodorus Metochita 
apud Sathas Msaaioav. §ipX. 1 p. q&: yiccl fiiiv hi xal &tta t^ 
&vSqI (Euclidi) nQoas^BlQyaatoci, djttMoc ts xofl wxtoTCtQiiLic lial 
dedoiiiva wxl tcc ytSQl t&v %at* oifQavbv qfaivoiiivtov , ^aTCBQa) 
nQ^^vQci T^i^a tavta nal TCQoavXta t&v ivtbg &JC0QQi^t0Dv ts %aX 
&d'6t(ov &6tQovo^lag\ u. etiam eiusdem vnofivriiiati.aiiot p. 108 
Eiessling. 

Apud Arabes nuUum uestigium est Catoptricorum (Studien 
iiber Euklid p. 152); nam quae apud Alhazen inueniuntur pro- 
positiones similes (Schneider Eclogae phys. II p. 231, 233), 
aliunde habere potest (uelut ex Herone et Ptolemaeo), nec 
apud IJitellionem, quamquam multo plures propositiones similes 
habet (u. Schneider 1. c), similitudo eius modi est, ut e Cat- 
optricis nostris hausisse demonstrari possit; est enim multo 
diligentior et uerbosior. sed ut Opticorum, ita Catoptricorum 



*) Habet etiam Heliodorus cap. 11; idem cap. 13 ex Herone 
citat prop. 1. 



PBOLEOOMENA. LI 

interpretatio Latina ezstat e Graeco facta saeculo circiter Xm, 
cm titulus est Euclidis de speculis*). eius hosce codices 
noui: 

cod. Marcian. Lat. 382 s. XTTT fol. 252, cod. Florent. Cony. 
soppr. I V 80, Uindob. Lat. 6210 s. XIV fol. 88—96', Norimb. 
cent. V, 64 s. XV fol. 168^ — 170«, Amplon. Q 887 s. XIV 
fol. 42—44^ Dresd. Db 86 scr. a. 1410, Dresd. Db 86 s. XIV, 
de quo supra, cod. Musei Britann. Add. 17868, cod. Oxon. 
Coll. Corp. Chr. 261 et 288 fol. 166—167, cod. Cantabr. Uni- 
uersit. Mm m, 11. e Db 86 quaedam excerpsi, unde adparet, 
interpreti exemplar codici Paris. 2448 simillimum ad manus 
fuisse, quod intelleget, qui scripturas in&a adlatas cum col- 
latione codicis Parisini comparare uoluerit: 

p. 286, 1 rectum uisum esse, cuius media terminos recte 
contmuant, p. 290, 6 equalem faciens angulum, 10 positus uero 
est et ez angulus, 18 conueniet autem, p. 296, 17 trahantur, 
p. 810, 6 iaceantque, p. 312, 18 quare erit ae et he^ p. 814, 17 
in directo eius que est de^ p. 830, 22 ag partes, p. 840, 13 re- 
fractus da cadit, 17 et equales periferias deprehendentes (sed 
p. 298, 13 manifestum uero, p. 332, 5 diametrum, p. 838, 18 om- 
nino, p. 340, 21 et ab aliis radii Be ^ z deg et zta); p. 802, 26 
habet: igitur expulsis uisibus. 

itaque de Graeco fonte dubitari non potest, sed ut in 
Opticis, ita hic quoque interpres alium quoque habuit; nam 
in demonstrationibus saepe a Graecis differt et in mg. tantum 
iis similia praebet. speciminis causa adfero propp. 1 — 8: in 
planis speculis et conuexis et concauis uisus in equalibus 
angulis reuertitur. esto oculus &, speculum planum ag^ uisus- 
que ab oculo feratur 6fc et reuertatur super d. dico, quod 
anguli reflexionum sunt equales, qui scilicet continentur sub 
speculo et radio emisso et radio reflexo. trahantur enim per- 
pendiculares a dhg super ag^ et erunt ad Jcg hk trianguli 
similes. latera enim proportionalia sunt per elementa posita 
et anguli contenti sub proportionalibus lateribus equales. quare 
trianguli sunt equianguli. quare k anguli equales. esto uero 
speculum conuexum agk uisusque kh et reuertatur super d. 



*) Aliud opus sub hoc titulo peruulgatum medio aeuo com- 
memorat Bose, Anecd. n p. 291. exstat etiam in cod. Magliab. 
XI, 80 et XI, 66, Dresd. Db 86 f. 274«, Paris. sup^l. Gx. ^<5^% 
f. 179^ 



LE PROLEGOMENA. 

si igitur intelligainus speculum planum contingentem circulum 
in puncto k^ facient idem radii scilicet hk dk angulos cquales 
cum speculo plano. idem enim punctus adhuc est reflexionis, 
qui prius, sed anguli contingentie sunt equales in eodem cir- 
culo. quare totus angulus toti angulo. idem est in speculo 
concauo. supposito enim speculo plano, cum anguli oontingentie 
sint equales, erunt et anguli portionum*). totalis enim totali 
equalis est. et ex hoc manifestum est, quod non nisi in unico 
pimcto possibile est fieri reflexionem et in quolibet speculo, 
ut scilicet uideatur eadem res ab oculo in eodem situ**) ma- 
nente. hoc tamen satis constat per decimam. 

qualitercunque speculo inciderit uisus equales faciens angu- 
los, is per se ipsum reuertitur. hoc manifestum est. si enim 
non reuerteretur per se ipsum quacunque parte facta reflexione, 
faceret partem equalem toti per ypothesim et primam pro- 
positionem quocunque existente speculo. 

qualitercunque speculo adueniens uisus inequales facit 
angulos, is nec per se ipsum nec super minorem angulum 
reuertitur. si enim per se ipsum, faceret angulos reflexionis 
equales contra ypothesim. si uero super minorem angulum, 
faceret per primam partem maiorem toto, quia equalem maiori 
suo toto. 

ad propp. 2 — 3 in mg. adsciiptae demonstrationes genuinae, 
sicut ad propp. 4, 20, quae a Graecis discrepant, ut propp. 6, 
21 — 23, 28; cum Graecis concordant definitiones et propp. 6 
—18, 27, 30, 31 et magna ex parte 19, 24r— 26 (p. 298, 26 fan- 
tasia). scholia nonnulla adsunt, uelut ad prop. 7: nota, quod 
in quibusdam libris scribuntur 16 et 17 et 18 ante istam 7 
propositionem. 

in cod. Torun. R IV^ 2 p. 68 huius interpretationis duae 
propositiones ultimae leguntur solae, quas subiungam: 

possibile est speculum construi et in eodem apparere plures 
facies, has quidem maiores, illas uero minores, has quidem 
propinquius, illas uero longius, et hic quidem dextras, illic 
aero sinistras. esto enim planum ah. ergo in eo fiunt utique 
conuexa specula ut aog et trk, concaua uero ut gde et zit^ 
plana ut ez. posita uero facie***) apparent quidem a speculis 
planis equalia ydola et equaliter distantia, a conuexis uero 



*) corr. ex portionem. **) in ras. ***) seq. lac. 



PROLEGOMENA. Lin 

minora et minus distantia, a concauis omnino magnitudine*), 
quemadmodum demonstratum est. 

ex concauis speculis ad solem positis ignem accendere. 
esto concauum speculum abg, sol uero zde, centrum uero 
speculi t, et ab aliquo puncto solis ut d coniugata super cen- 
trum dJcb recta trahatur. incidat autem dg radius et re£ringa- 
tur super k. non autem reMngetur super centrum t. angulus 
enim igd**)^ qui est ad circumferentiam , minor est angulo 
semicirculi. et esto ab periferia equalis bg periferie, et incidat 
alius radius da. manifestum est igitur, quod da refractus 
ueniet super k; equalium enim periferiarum eiusdem circuli 
equales sunt anguli. similiter autem demonstrabitur, quod 
omnes radii a puncto d speculo incidentes inequales periferias 
deprehendentes circa rectam tb refracti coincident in aliquod 
idem punctum recte tb. esto rursus concauum speculum bag, 
sol uero dze, et ab aliquo puncto solis ut e per t centrum 
esto radius etb^ et ab aliis scilicet d z sint radii dtg et zta. 
et quoniam omnes radii transeuntes per centrum faciunt equales 
angulos ad periferiam, prof) eo scilicetf) quod faciunt angulos 
semicirculorum, onmes refe-inguntur super se ipsos ad centrum. 
hiis ergo radiis per concursus ad eandem partem calefactis 
ignis accenditur. 

eandem citare uidetur Eogerus Baco Op. mai. p. 51: et 
ideo oportet, ut speculo concauo ad solem posito ignis accen- 
datur, sicut dicit ultima propositio libri de speculis; p. 808: 
speculo concauo ad solem posito ignis accenditur, ut dicit ultima 
propositio de speculis, scilicet in puncto axis, ad quem reflec- 
tuntur omnes radii circumferentie unius circuli; unde si stupa 
uel aliud combustibile apponatur, sole fortiter radiante com- 
buri potest in puncto illo. sed Euclides de speculis ei aliud 
opus est; u. p. 309: docet enim Euclides in 83 propositione de 
speculis sic figurari speculum, ut congregatio radiorum flat 
ante et retro, p. 310: et sicut dicit Euclides libro de speculis 
et probatur in 7 propositione, figura lucis est maior quam 
foramen; cfr. p. 306: et hanc probationem eandem affert 
Euclides ad 5 propositionem sui libri (agitur de Catoptr. 1), 
p. 330: Euclides docet figurare speculum, quod comburat ante 
et retro. nec Albertus Magnus nostrum opus ob oculos ha- 



*) comp. dubium. **) gd post ras., * add. mg. 

t) comp. dub. 



LIV PROLEGOMENA. 

buisse uidetur, cum dicit Meteor. n p. 127: adhuc autem^ 
sicut dicit Euclides, speculum non tantum manifestat imaginem 
rei, sed etiam distantiam eius a speculo, quia res, quae longe 
distat a speculo, uidetur esse in profundum speculi ad tantam 
distantiam, ad quantam distat a superficie speculi (cfr. Uin- 
centius Bellouac. Specul. nat. n, 80: quoniam in speculo non 
resultat forma tantummodo, sed etiam distantia, quae est inter 
adspicientem et speculum) et De sensu et sensato Y p. 11 r 
taliter potest moueri eleuando et deprimendo speculum, quod 
uidens uidebit speculum et tamen non uidebit se ipsum in. 
speculo, sicut demonstratum est ab Euclide in prospectiuis. 
de Uincentio Bellouacensi res incerta est; cfr. Spec. nat. n, 77 : 
ab Euclide inuenitur probatum, quod reflectio luminis semper 
fit ad pares angulos uel in se ipsum; ad pares quidem angulo» 
fit, si radius ex obliquo ueniens est ad superficiem speculi, in 
se ipsum autem, si perpendiculariter (Catoptr. 1 — 2); n, 81: 
non solum uero adparet dextra sinistra et e contra in speculis 
conuezis (Gatoptr. 20), sed etiam in planis (Catoptr. 19), non 
tamen ex causa, quam in libro de speculis ponit Euclides, 
uidelicet eo quod uideamus per lineas radiales ab oculis egre* 
dientes (Catoptr. p. 286, 1). sed uidetur tamen nostrum opus 
respicere. 

sine ullo dubio respicitur in tractatu de speculis apud 
Combach, Baconis perspect. p. 168: ex concauis speculis ad 
solem positis ignis accenditur. haec ultima propositio libri de 
speculis commimibus sic demonstratur ibidem (sequitur demon- 
stratio, sed amplior), et a lohanne Peckham Persp. commun. 
II, 17: hinc est, quod a speculis concauis sphaericis ad solem 
positis ignis accenditur (cfr. 11, 66), et 11, 60: in speculis con- 
cauis res nunc conuersas nunc euersas apparere. hanc demon- 
strauit Euclides de speculis (Catoptr. 11 — 12; sequitur demon- 
stratio a Graeca diuersa); 11, 62 uero: in speculis concauis ex 
diuersitate situum quaedam apparere recta quaedam curua 
quaedam conuexa . . . diffiise demonstratur libro sexto cap. YII 
Alhacen, Euclides autem tantum apparentis curuitatis meminit^ 
ex alio opere sunt. 

Georgius UaUa De expet. et fug. rebus XV, 2 etiam e 
Catoptricis quaedam transtulit (Neue Jahrb. Suppl. XII p. 396). 
eum et Zambertum iisdem codicibus , quibus in Opticis , usos 
esse, consentaneum est; cfr. p. 286, 8 vijjog] ^ovg Monac. 361, 
speetcmHs fastigium Ualla, aspecfti fasUgii Zambertus, p. 288, 10 



PBOLEGOMENA. LY 

AKV] AK Monac, alc Ualla, alcc Zambertus, p. 314, 1 A (alt.)] 
AE Monac., ae Ualla, a Zambertus, p. 330, 11 &vu%ko)yLiv7i — 
12 ^|st] om. Monac, Ualla, Zambertus. sed Ualla scholia 2, 
3, 4, 5, 7, 8 habet, quae in Monac non exstant. 

Pena in Gatoptricis quoque cod. Paris. 2350 habuit; u. 
p. 330, 11 &va%Xa}ii,iv7i — 12 ^S«t] om. 2350, post B© lin. 12 
add. sl yicQ dvvatov Uergetius in mg., Pena. memorabile est, 
Uergetium hic in interpolando codicem Paris. 2448 usurpasse*), 
uelut p. 301, 1 inde addidit mg §oi&'og cpccivstoci; cfi*. scripturae 
Penae p. 286, 20 Q-s&^rnia a\ p. 288, 9 iv roo xvproo ivdittQtp^ 
19 iv t& HoiXq) ivdTtZQcp, p. 292, 20 iv tm yivgt^ iv67ttq(p^ p. 294, 22 
ifftfto; iv t^ jtSQupSQsloc , p. 314, 6 aviipocciv] fSvii^oXrjv^ &'jt6'\ om. 

Dasypodius prius totum opus ediderat Argentorati 1557 
codice Marciano 301 eiusue apographo usus ; nam interpolationes 
eius habet (u. Studien iiber Euklid p. 148 — 150; p. 288, 4 ya^ 
habet, p. 288, 15 VMK, sed ietl lin. 6 et &XXcc dri forco lin. 9 
non habet). postea a. 1571 propositiones solas repetiuit iam 
editionem Penae secutus (Studien p. 149 not.). 

Gregorius in Gatoptricis nuUum codicem nominat, sed a 
Pena solo pendet. eum sequitur Schneider Eclog. phys. I 
p. 391 — 394, ubi Gatoptricorum quoque propositiones enumerat. 



*) Uestigium codicis m deprehendi p. 328, 20, ubi ad iii- 
niaji adscripsit in mg. Uergetius y^. i%ts^^, quod in mg. 
retinuit cod. Paris. 2468, nec recepit Pena {tsd-fj m, Dasypodius). 



EUCLIDIS OPTICA. 



JSuclideB, odd, Heiberg et Menge. VU. 



EUCLIDIS OPTICA. 



Eaclides, edfd. Heiberg' et Menge. VU. 



'Oqoi. 

1. ^TtcokbC^^^g) t&s «^ro tov bfi(iatos i^ayofisvag 
sid^eias yQaiificts (pBQB^^^av Sid^trjfia (iByBd^&v (iBydXcjv. 

2. xal tb [[ihv^ {)7cb t&v ^bcov 7Cbqibx6[ibvov ffxVi*^ 
6 Blvat XGtvov tiiv xoQVfpiiv fihv ix^vta iv tp ^iiiiatL ri)v 

Sh pd^LV TCQbs tOLS JtBQa6V t&V 6QCJ[livC3V, 

3. xal dQctdd^aL nhv tavta^ TCQbs ct ctv al '6i)Bis 

TCQOdTCLTCtCJdL^ (l"^ bQCC^d^aC Sb^ TCQbs ct CtV (l"^ 7CQ06" 
7CL7CtGi6LV aC ^if.BlS. 

10 4. xal ta (ihv iTcb ilbC^ovos ycnvCas bQ^fiBva fiBC- 
tfiva (paCvB^d^aij t& Sh ijcb ikdttovos ikdttova^ t6a S% 
t& iTcb t^cnv ycavLGiv dQwnBva. 

5. xal td fihv i>scb iiBtBcoQotBQGiv dxtCveav 6Q(QfiBva 
fiBtBG)Q6tBQa (paCvBdd^aL^ tcc Sh ijcb tojCBivotiQov ta- 

15 7CBLv6tBQa. 

6. xal 6iioCg)s 'C^ l^hv 'bicb SB^LcatiQov dxtCvcjv 
^Q&ikBva SBi,Ld)tBQa (paCvB^d^aLj td Sh ^Tcb aQL^tBQcati- 

QC3V &QL6tBQG}tBQa. 

7. td Sh ^Tcb 7cXbl6vg)v ycavL&v 6QW[iBva &KQL^i6tB- 

20 Qov ipaCvB^d^aL. 

a. 

OiShv tcbv 6QG)fiivGiv a[ia 5kov ^QcttaL. 
i6tG) yccQ 6QG)fiBv6v tL tb A^^ bii[ia Sh l6tG) tb B^ 
&q)' oi 7CQ067CL7CtitG)6av '6^BLS al BA^ BF^ BK^ JBz/. 

■ 

1. ^yXbLSov ditxi^ol oQoi VVat.Bvm; E^Xeidov d^rrixa. 
Zqoi rovtoDv Vat.^ numeros om. codd. 4. {isv] deleo; iibc v, 



Ponatur, ab oculo rectas ductas lineas ferri spatio 
magnitudinum inmensarum; et sub uisibus contentam 
iiguram conum esse uerticem quidem in oculo haben- 
tem, basim uero ad terminos conspectorum; et ea 
quidem uideri^ ad quae uisus inciderit^ non autem 5 
uideri^ ad quae non inciderit uisus; et sub maiori 
quidem angulo uisa maiora apparere, sub uero minori 
minora^ aequalia autem sub aequalibus angulis uisa; 
et sub eleuatioribus radiis uisa eleuatiora apparere, 
sub humilioribus uero humiliora; et similiter sub 10 
dexterioribus quidem radiis uisa dexteriora apparere, 
sub sinistrioribus uero sinistriora; sub pluribus autem 
uisa angulis perspicacius uideri. [omnes uisus aeque- 
ueloces. non sub quocunque angulo rem uideri.] 

Nullum uisorum simul uidetur totum. 15 

esto enim uisum quidem ad, oculus uero esto fe, 
a quo incidant uisus ha, hg, hk, hd. igitur quoniam 

9. radiis] M, angulis Z>. 13. omnes — 14. uideri] om. L; 

quidam libri habent ista duo principia et quidam non D mg. 

16. enim] ML, autem Z>. esto(aZ*.)] L, om, D. 17. 
hTc\ ML, hfk D. 

fio" B et Vat., corr. m. 2. 6. pcofteVcov v. 7. av] om. 

Vat.^m. 8. TtQOGitlnrmGi] nQOGTciitxovtsi Vat.^m. tcqog- 

nintcaCLv] ytQOCTtinroaGL v. 10. dQdnsva v. 11. iXdrrovcc] 

ildaaova VVat.v. 12. 'bn^] &n6 Vat. 24. nQoaninrera} v 
et Vat., sed corr. ai 6^st,g al Vat.v. 



EUCLIDIS OPTICA. 




oi)Kovv^ iTCsl iv Sia6tT^iiatv (pigovtai al 7tQo67tL7Ctov6ac 

S^fig, OVX &V 7CQ067CC7ttOLBV 6VV- y^ T JT /1 

exstg 7CQbg tb Azf' &6tE yivotvto av 
xal Katic tb A/1 8ia6xi{yi>ata^ 7CQbg a 
5 a[ btpevg oi) 7CQ067CB6ovvtai, oix aQa 
6(p%"ifl6Btac oAoi/ «fia xb AA, SoxBt 
Sh bQa^d^ai a[ia t&v 'oipBOJv ta%i) 

7CaQa<pBQ0[livGiV. 

10 Tg)v t6(ov iiByBd^oiv iv SLa^ti^iiatL xBL[iivG}v tic 
Byyvov xBLHBva aKQi^i^XBQOv bQatai, 

i6tc3 (iiifia fihv ro J5, 6Q(h[iBva Sb tb Fzf xal tb 
^ KA^ XQ^l Sb voblv aitd^ l6a xal ^caQccklrjXa^ iyyLOv Sl 

B6t(0 tb F^5 TCal 7CQ067tL7Ctit(O6aV 

16 'o^BLg at BT^ JBz/, BK^ BA. oi) yccQ 

av BikoLfiBV^ G}g al ascb tov Siifiatog 

TCQbg ro KA 7CQo67CL7ttov6aL HtlfBLg 

Sloc tG)v r^ A 6rillBLG)V ikBv6ovtaL, 

fi yocQ tQLy6vov rov BAAKFB f^ 
20 KA [iBL^(ov av ^v trjg FA' i)7c6- 

%BLtaL S\ xal t6Yi. oi^^aovv ro FA V7cb tcIblovcdv 'oiI^bgjv 

bQoxaL ^7CBQ tb KA, axQLfii^tBQov aQa cpaviq6BtaL tb 

TA tov KA' tic yccQ V7cb 7cIbl6vg)v ycovLav oQCj^Bva 

axQLfii^tBQOv (paCvBtaL, 

25 y. 

"EKa6tov tG)v 6QG)^ivG)v Ix^L tL firjxog aTCo^trJiiatog^ 
oi yBv6^Bvov oixitL oQcctaL. 

B6tG) yaQ oftfia iiiv ro J5, 6q6iibvov Sb tb FA. 
(prjfil dij, ort ro FA bv tLVL d7C06t7lfiatL yBv6^Bvov 

3. yivoivto Vat., yivsto v. 8. TtSQKpsQOnsvcav m. 11. 
Myysiov V, corr. in. 1; item lin. 13. 12. oQmnsva] corr. ex 




EUCLIDIS OPTICA. 5 

in distantia feruntur incidentes nisiis, non qnidem 
incidunt continue ad ad. quare fient et in ac? spatio, 
ad quae uisus non incident. non ergo uidebitur simul 
totum ad. uidetur autem uideri simul uisibus uelo- 
citer transportatis. 5 

Aequalium magnitudinum in distantia iacentium 
propius iacentia perspicacius uidentur. 

esto oculus quidem 6, uisa uero gd et lcl, oportet 
autem intelligere ea aequalia et parallela. propius 
uero sit gd. et incidant uisus hg^ hdy bJc^ 6?; non lo 
enim dicemus, quod ab oculo a,d Jcl accidentes uisus 
per g^ d puncta ueniant. trigoni enim hdllcgb recta 
Jcl maior utique erit recta gd; ponitur quidem aequalis. 
igitur gd sub pluribus uisibus uidetur quam fcZ; per- 
spicacius igitur gd quam M] sub pluribus enim angulis 15 
uisa perspicacius uidentur. 

Unumquodque uisorum habet longitudinem ^patii, 
quo facto non iam uidetur. 

esto enim oculus 6, res autem uisa gd [sub minimo 
angulo uisui determinato]. dico, quod gd in aliquo 20 



3. incident] L, incidunt Z>. 5. transportatis] L, trans- 
positis D. S. k^ L, ki D, et sic per totam prop. 10. in- 
cidant] L, inciduntD, incident M. 11. dicemus] L, onmes I>. 
19. sub — 20. determinato] D, om. L. 



oQoaiisvov m. 1 V, dQto^svov Bv et Vat., sed corr. m. 2. 16. 
stTtofisv VVaiW. 19. Post yccQ add. av m. 2 Vat. 20. 

^dKSitcci] corr. ex vitoTislGd^G) m. 2 V, ^nov.siG^oi v (o corr. 
ex a^) et Vat., corr. m. 2. 21. 8s] om. Vat.v. 26. icito- 

ari^lLatcc v. 27. ysv6iisvov] corr. ex ysvo\LivQV tel. ^ V^^ai^.^ 

ysvofisvov v. 28. B] e corr. Vat. 



EUCLIDIS OPTICA. 




oixsn 6Qad">]6sta(„ ysysvjlfi^Gi ykq 
xh T/1 iv r& iiera^i) Sia^rTJfiatL t&v 
btpsiov^ i(p* oi tb K. oixovv TCQog 
tb K oiSsiiLa ta)v &%b rov J5 'o^eov 
5 nQ067ts6ELtaf TtQos o Ss al bil^SLg oi 

7CQ067CL7CtOV6LV^ ixStVO O^X ^QCCtaL. 

ixa6tov &Qa t&v bQco^svcDV i%SL rt ftij- 

xog &7Co6t7lfiatog^ oi ysvdfisvov oixstL 

bQataL, 
10 *'. 

Tg)v t6(ov SLa6trjndta)v xal i7cl t7\g a^btrig sid^siag 
'6vt€ov ta ix TcksCovog SLa^f^fiatog bQ^fisva ildttova 
ipaCvstaL, 

i6ta) t6a SLa6ti^iiata i7cl (iLctg sv^sCag tct AB^ BF^ 

16 JTz/, xal avTjx^^ ^Qog bQd^ag fi AE^ iq>^ f^g xsC^d^o 

^lifia tb E. Xsyoj^ oxl fist^ov q>a- A B V /1 

vrJ6staL tb (ihv AB tov J5F, ro 

Ss BF tOV Fzf. 7CQ067CL7CtitG)6aV 

y&Q axttvsg al EB^ EF^ Ezf^ xal 

20 ^xd^cj SLa tov B 6rjfisCov t^ FE 
eid^sCa TCaQdXXrjXog ij BZ. t6ri aQa 
i6tlv ii AZ tfj ZE. i7Csl yaQ tQL- 
yd)vov tov AEF 7CaQ& (iCav tcbv 
7cXsvqS)v tijv FE fjxtaL ^bd^sta ^ 

25 J5Z, i6tLV aQa xaC^ Sg fi FB TCQbg BA^ ii EZ TCQbg ZA. 
t6ri &Qa i6tlv rj AZ^ djg stQrjtaL^ tfj ZE. ^sC^g)v S^ 
TcksvQdc fi BZ trjg ZA' ^sC^gjv &Qa xal tfjg ZE. (isC^g)v 
&Qa xal yovCa ii V7cb ZEB yovCag tfjg i^cb ZBE. xal 
fi i7cb ZBE tfj i7cb BET H^rj' xal fj i^cb ZEB &Qa 

3. ^qp'] &q)' Vat. 6. TCQoanlittovGi v. 8. yfvoftfvovl 

cojT. ex yspo/isvov m. 2 V, ysvoiLevov BVat.v. 14. Ante im 




EUCLIDIS OPTICA. 7 

spatio factum non iam uidebitur. fiat enim in inter- 
medio spatio uisuum^ in quo Tc, igitur ad Tc nullus 
ab 6 uisuum accidet. ad quod uero uisus non in- 
cidunt, illud non uidebitur. unumquodque ergo uiso- 
rum habet longitudinem spatii, quo facto iam non 5 
uidebitur. 

Aequalium spatiorum et super eandem rectam 
existentium e maiori spatio uisa ndnora apparent. 

sirit aequaKa spatia super eandem rectam db, hg, gd, 
trahaturque perpendicularis ae, in quibus iaceat ocu- 10 
lus e. dico, quod maior apparebit db quidem quam 
hg et hg quam gd. accidant enim radii eh, eg, ed, 
et trahatur per punctum h rectae ge parallela hz. 
aequaJis ergo az recta rectae ez. quoniam enim tri- 
goni aeg circa unum laterum ge ducta est recta hz, 16 
est igitur quod sicut hg ad ha, ita ez ad za. aequalis 
ergo az, ut dictum est, ze. maius uero latus hz 
quam za. aequalis uero za ze. maior igitur angulus 
zeh angulo zhe. angulus quoque zhe angulo heg aequa- 

1. in] X, om. D. 4. ergo] X, igitur Z>. 5. iam] X, 

07W. D. 16. quod] q, D (que.^). 

del. ra m. 2. V. 16. xa/] om. v. -4E] E in ras. V. 

18. T^ NA V. 19. EP] EB v. 20. rjl e corr. Vat. 

VE^l EF m. 22. itsrLv] om. BVat.v. iTCsi'] corr. ex 

inl V. 23. AET] KET y. nocQa\ TtSQl v, n' Vat. 25. 
FB] BF BVat.v. BA] tiiv BA BVat.v. ZA] ttiv ZA 

BVat.v. 26. iexiv] om. Vat. Dein del. rj ZE ^s/fcoi; B. 

8i\ corr. ex di} V, ovv BVat.v. 27. rfig (pr.)] x^ BVat.v. 

Dem add. tcri 8\ i} ZA rj ZE BVat. v. ^ajov (pr.) — ZEl 
om. Vaiv. (LsiSav (pr.) — 28. &qcc] in ras. V. 28. ZEBJ 
E e corr. B, ZB v. y(oviag] yavicc V. ,rfjg] m. 2 ex r{ V. 

ZBE] E in ras. V. xal rf] ij 8s m, et in ras. V. 29. 
ZBE] e corr. Vat., ZEB v. BET] BEN Bn. T.^*^ 

EB V. 




8 EUCLIDIS OPTICA. 

tfjs 'l>^o FEB yfovias fisi^Giv icrCv. (isl^(ov &Qa 6q>%"i/i- 

6srav ii AB rTJg BF. %dXtv dfiOLGis ^otv Sia rot) V 

6rjiisi(yv rrj ^E JtaQccllrjlog a%%^^ iisv^ojv 6<pd"^6sraL 

il BF rijg Tz/. 
5 s . 

Tic l6a iisysd^rj avi6ov SLs6rrjx6ra avv6a tpaCvsrai^ 

xal fist^ov asl rb lyyiov xsCiisvov rov 'ofiiiarog. 
s6raj Svo i'6a fisysd^rj ra AB^ Pz/, 

Sfifta d^ i6r(o ro E^ acp^ ov avL6ov 
10 Sts^rrjxdrco^ xal i6r(o Syyvov ro AB. 

Isyco^ orv nst^ov (pavr]6srav ro AB. 

7CQo67tL7Crsrio6av axrtvsg aC AE^ EB^ 

EF^ Ezf. ixsl oiv rct: i^o [isv^6v(ov 

yiovL&v 6Q(oiisva ^sC^ova (paCvsrat^ 
15 fisC^(ov Sl ycovCa i\ viio AEB rr\g 

{)7cb FEd^ (isC^(ov aQa (pav/j^srav xal ii AB rfjg JTz/. 

g'. 

Ta TtaQakkrika rcbv SLa6rriiidr(ov ii, aTCO^riJiiarog 
6Q(OfAsva dvL607ckarri (paCvsraL. 

20 i6r(o Sijo TCaQdXkrjka [isysd^rj rd AB^ JTz/, ti^^a Sl 
i'6r(o ro E. kiyco^ ort rd AB^ JTz/ dvL607ckarrj (paC- 
vsraL^ Tcal [ist^ov dsl ro SyyLOv SLd^rrj^a roi> 7Coqq(o- 
rsQOv. 7CQ067CL7Criraj6av dxrtvsg at EB^ EZj ES^ E^y 
EH^ EKj xal i7Cstsvx%co6av svd^staL a[ J5z/, Zff, SK. 

25 i^sl ovv [isC^cov i6rlv ii V7cb BE^ ycovCa rrjg i^cb 
ZEH ycDvCag^ [isC^(ov aQa xal ii B^ rfjg ZH (paCvsrat. 

1. TEB] BEF BVat.v, EFB Vat.*m. 2. %(kv] xat m. 

3. icxQ^v^^ in ras. V. 6. &vi6ov] corr. ex &vl60iv v. 7. 

%yyBiov V, corr. m. 1, ut lin. 10. Hnarog v. 12. -4E] EA 
BVat.v. 15. AEB] r&v AEB BVat.v et V, sed corr. 

16. &Qa] om. m. 22. ^yysiov V, sed. corr. 23. TtQoa- 

niTtriro) Bv. EJ] EK By. 24. EK] EJ Bv. 25. iari v. 



EUCLIDIS OPTICA. 9 

lis. iergo beg angulo zeh angulus maior est. maius 
ergo uidebitur ah quam hg, rursum simiKter si per 
punctum g rectae de parailela ducatur, maius uidebitur 
hg quam gd. 

Aequales quantitates inaequaliter distantes in- 6 
aequales apparent et maior semper propinquius iacens 
oculo. 

sint duae aequales magnitudines ah, gd, oculus 
uero sit e, a quo inaequaliter distent, sitque propin- 
quius ah. dico, quod maius apparebit ah, accidant 10 
enim radii ea, eh et eg et ed, quoniam ergo sub 
maioribus angulis uisa maiora apparent^ maior autem 
angulus aeh quam ged^ maius ergo apparebit ah 
quam gd. 

Aequidistantia spatiorum e distantia uisa inaequa- 15 
lis magnitudinis apparent. 

sint duae parallelae quanti- 
tates ahj gd, oculus autem 
sit e. dico, quod ah et gd 
inaequalis latitudinLS appa- 20 
rent, et maius apparebit sem- 
perpropinquius spatium quam 
remotius. accidant radii eh 
et ez et et, eJc, el et ed, et 
coniungantur hd, z\ tlc. quo- 25 
niam ergo maior est hed angulus angulo ^ely maior 
ergo hd quidem linea quam zl apparet. rursum 

16. magnitudinis] scr. latitudinis. "iS. al\ e wcv. T> 

(l semper corr. in hac prop.). 




10 



EUCLIDI8 OPTICA. 



xdXtv iatl (tsii<ov ^ i!xb ZEH ymvCa z^g ^h &EK 

ymvias, fiBi^mv &(fa xal ij ZH r^g @K fpaivtxai. (isl^ov 

«pa t6 (liv BJ dtdfftrifta tov ZH, ib Sl ZH zov &K. 

Qvxirt oiv 6^&~^0etai JcaqdXXrjka 8vza tic SiaStiifiata 
6 ia' fffijs, iXX' &vieo«la.tfi. 

htX zSsv iv iiets(oQe> XEi(iivo)v StaOzrjHKXiav xa&i- 

ie&es &itb roi) A etjftiiov inl tb -Ozoxsifievov ixixeSov 

xtt^stos ■^ AB, xal iffztoOav ^ j^ 

sapaAAijiot aC AS, KN, &M. 
10 Xdym, ozi xal ovtmg «viffo- 

aXazri ipaCvstat t^ F^, EZ 

ftsyi&Tj. ^xfrra xdd^ezog &itb 

tov B inl tijv AS ^ BP, 

xal ix^efilija9m ■^ BP ial 
16 tb 0, xal ngogniJizetmeav &x- 

ttveg at AA, AK, A&, A^, 

AN, AM, xal i^s^iix^foaav 

al AP, AII, AO. ixsl o5v 

Aab (uz£a(fozi(fov ffijficCou rou A ixl t^v PS ini^evxtat 
20 ttg ev^sta ■^ AP, ■fj AP apa ial f^v Pff xd9et6s 

istiv, xal i) AO in\ zi^v OM, xal ij AII ijtl t^v HN. 

^(fQ^oydivia &Qa iozl t^ APIS, AIIN, AOM zQtyiava. 

insl o^v 6^&oyiovtd iiSti, xaC ietiv ij (liv IIN tfi PS 

tetj, ^ 6h HA zijg AP (ieC%mv, [uCtmv S(fa ymvCa ij 
S6 -bab SAP r^s *re6 HAN. ftcrgov ^fpa xa\ 6<p»ijeezat 

tb PS tov IIN. bfioCms xal tb PA toi) HK [leitov. 

SXov &(fa tb AS SAou roC KN 6q>d^aszai ftcf^oy. 

ivieoTtXazTj /i(fa xal oikms &ip9''^estai td fisyi9ij. 



U' 



8. St^STitta V 



6._ r VVat.Bin 
12. ^d&CTos] ut raa 



. AS] 



EUCLIDIS OPTICA. 11 

quoniam maior 0el angulus quam teJc angulus, maior 
ergo ^l quam tik apparet. maius ergo bd spatium 
quam isl et maius 0I quam tlc. non iam ergo uidebun- 
tur parallela existentia spatia aequaliter^ sed uidebuntur 
inaequalis latitudinis. 5 

in eleuato iacentibus spatiis demittatur ab a puncto 
super subiacens planum catetus ah. suntque parallelae 
Ix, Jcn, tm, dico, quoniam et sic inaequalis latitudinis 
apparent gl et xe magnitudines. trahatur enim cathe- 
tus a puncto 6 super Ix br, et educatur br super o, 10 
et accidant radii al, ah^ at, ax, an, am. coniungantur 
aVj apy ao. quoniam ergo ab eleuato puncto a super 
Ix coniuncta est recta ar, igitur ar super Ix cathetus 
ei ao super tm et ap super pn. ortogonii ergo sunt 
arx et apn et aom trigonii. quoniam ortogonii sunt, 15 
et est quidem pn ei quae est rx aequaKs, pa autem 
quam ar maior, maior ergo angulus rax angulo pan, 
maius ergo uidebitur rx quam pn. simiKter autem 
et Ir quam pJc. totum ergo Ix toto Jcn uidebitur 
maius. inaequalis ergo latitudinis et sic uidebuntur 20 
magnitudines. 



10. 0] pro 0, ut uidetwr, semper c hab. D. 13. Ix] mpra 
scr. JD. 



Tl}v m. l^. AO] A@ ^\. 19. ILBtBOiqOV Bv. iTte^SVTltCCL 

— 20. P^] om. m. 21. icti m. IIN] NTI m. 22. 

6Q&oy&viov m. ra] to m. 23. iativ] iati v. 24. iisL- 

icav (pr.)] corr. ex iiigog V. 25. 11 AN] t&v UAN V. 

ftfiifov V. 26. PA] AP By. JTJf] KU m. ^L^lt^^i-vX 

om. Bv. Hoc Joco errore nihil e Yat. eB.o\»»m. 



12 



EUCLIDIS OPTICA. 




Tct: iTtl XYis ccdtrig avd^aCag Svra t6a fisyEd^rj fti^ 
iq)6^fig aXX7]XoLg tsd^ivra xal &vv6ov dLe^trjxdta tov 
9fi(iatog avi(5a (paivetai. 
5 i6t(D dxJo t6a (isyid^rj tcc AB^ Pz/ inl trjg aittrjg 
svd^siag tfjg A/l iiij i^ps^rjg «AAiJAotg '6vta xal avi6ov 
disdtrjxdta aTcb tov Sftfiarog 
rov E^ xal 7CQ06%L%titG)6av 
axttvsg al EA^ EA^ xal 

10 l6tcj fisCtcDV ri EA tfig EJ. 
Xiyio^ Srt ii FA tfjg AB 
fiSL^cov <pav7^6staL. nQ06- 
7ei7ttitaj6av axttvsg a[ EB^ 
EF^ xal TtSQLysyQccq^d^o tcsqI 

15 ro AEzjf tQCycovov xixlog 
6 AEA. xal tcqo^sx^s- 
pXri^d^cj^av tatg EB^ EF sid^sCavg svd^stav av BZy 
JTff, xal avs6tdtco6av aith t&v B^ F 6rinsCc3v TCQog^ 
dQd^d^g ycDvCag l'6at sid^siai aC BS^ FK. i6tv 81 C6rj 

20 fj AB tfi JTz/, «AAd xal ycnvCa ii 'bTcb AB0 tf] ^Tcb^ 
/dFK i6tLV l'6rj. xal TCSQvq^iQSva aQa fi AS TCSQLfpsQsCa 
tfi ^K i6tLV C6ri. ri KzJ aQa nsQL(piQSLa tfjg ZA 
TCSQLfpsQsCag [isC^cov i6tCv. tcoIIS aQa fj H^ tcsql- 
(piQSva tfjg ZA ^isC^cov^^i^tCv. &XX^ iicl (ihv tfjg ZA 

25 7CSQLq)SQsCag f] vtco AEZ yojvCa piprjxsv^ i^l 8^ tfjg- 
HA 7CSQvq>SQsCag r] vTcb HEA. fi aQa 'VTcb HEA 
ycovCa tfjg i)7cb AEZ (isC^gjv i6tCv. aAA' i)7cb iihv tfjg^ 
i)7cb AEZ ij AB ^kinstaL^ vTcb dh tfjg ifTcb HE^ 
i] FA. (isC^cjv &Qa ij FzJ tfjg AB q^aCvstaL. 

1. n n' VVat.Bvm. b. AB^AHy. 6. icXX^^Xtov BVat. v. 
&riaov} ttviGov dLdarrj^icc m. 9. EA} AE v. 10. iietSov Bv. 



EUCLIDIS OPTICA. 13 

In eadem recta existentes magnitudines aequales 
non deinceps ad inuicem positae et inaequaliter sub 
oculo distantes inaequales apparent. 

sint duae aequales magnitudines a6, gd m eadem 
recta ad non deinceps ad inuicem existentes et iri- 6 
aequaliter distantes ab oculo e, et accidant radii ea 
et edy sitque maior ea quam ed, dico, quoniam gd 
quam ah maius apparebit. accidant radii eh et eg, 
et describatur circa aed trigonum circulus aed, et 
adiiciantur eh et eg punctis rectae hz et gi, et sur- 10 
gant ab fe, g punctis perpendiculares ipsis rectae 
aequales ht ei gJc. est autem aequalis et ah ei quae 
est gd. sed et angulus aht angulo dgh aequalis est. 
et periferia igitur Jcd periferiae ta aequalis. itaque 
Jcd periferia 0a maior est. multo ergo id periferia 0a 16 
periferia maior est. sed super za periferiam iacet aes 
angulus et super id periferiam ied angulus. angulus 
ergo ied angulo ae0 maior. sed sub illo quidem qui 
est ae0 angulus ah uidetur, sub angulo uero eid ea 
quae est gd. maior ergo gd quam ah apparet. 20 

2. sub] scr. ab. 14. periferiae] cwr. ex pariferiae D. 

15. pariferia J), ut saepius. 16. sed — 18. maior] mg. J). 
19. eie?] scr. ied. 

12. iLSt^ov Bv. 14. EF] om. v. 15. hvxXos] comp. BVat.v. 

16. TtQoasTipepXrJGd-G) Bv. 17. ET] seq. ras. unius litt. V. 
Evd-stai] om. V. 18. Tif] F supra scr. m. 1 v. Scvsatccro} 

Bv et Vat., sed corr. B] om. v, corr. ex A m. 2 Vat. 19. 
Laai] l'6ai avralg Vm, a-urars l'(SaL BVat.v. B@, FK] S et 
K e corr. V. 20. ii (pr.)] >tal i] B. 21. /iTK] in ras. V; 
BFK m. i] A&] om. Bvm, m. 2 Vat. 22. dK] in ras. V. 
23. iLsitfov i6tl TtSQicpSQsiag BVat.v 24. Z^ (pr.)] ZA nsQL- 
(pSQsiag BVat.v. tfjg (alt.) et 25. TtSQKpSQsiag] trjv — tcsql- 

(pSQSiav VVat.\ ut lin. 26 sq. 27. i6ti v. 28. vito (tert.)| 
m. 2 Vat. 



14 



EUCLIDIS OPTICA. 




TSt t6ifi iisyadTi Tcal naQdlXrjXa avL6ov Sis^trjxdTa 

&7cb tov 6(iiiatos oix &vak6ya}g totg 8ia6t7l(ia6LV bQatai. 

i6t(o dvo (leysd^rj tic AB^ JTz/ &vl6ov SLS6trjK6ta 

6 &7tb rov fi(i[iatos tov E. Xsycj^ oti ovx s6tiv^ i)g 
(paCvstav i%ov^ dyg tb Fzf TCQbg tb AB^ ovtfog tb BE 
TCQbg tb E/d. 7CQ06- 
7Ci7Ctdt(X)6av yo^Q 
axttvsg al AE^ 

10 EF^ xal xivtQfp 
fisv tp E Sia- 
6t7l[iatL Sl tip 
E Z xxJxAov ys- 
yQ&(p^(o TCSQifpsQSva ii HZ0. iicsl ovv ro EZF 

15 tQvyiovov tov EZH to^iiiog [ist^6v i6tiv^ tb S% EZ/I 
tQiymvov rov EZ® tofiitog llatt6v i6tLVj tb EZF 
&Qa tQLycovov TCQbg tbv EZH tofiia fisC^ova X6yov sxsc 
^TCSQ ro EZ^ tQLycovov TCQbg tbv EZS tofiia. xal 
ivaXkai, tb EZF tQLycjvov TCQbg tb EZzf tQiytovov 

10 fisL^ova l6yov i^SL i^TCSQ 6 EZH to[isvg TCQbg tbv 
EZS tofiia^ xal ^vvd^ivtc ro EF^ tQLycovov TCQbg tb 
EZ/d tQLycovov [iSL^ova l6yov i^sL V^tcsq 6 EH& to- 
fisvg JCQbg tbv EZS to^iia. aAA' d)g ro E^F JCQbg 
tb EZ^ tQLycjvov^ ovt(og ri JTz/ TCQbg ri^v ^Z. i^ Ss 

26 Pz/ t fj AB i6tLv l'6rj^ xal d)g ii AB TCQbg tijv ^Z, 
rj BE TCQbg tijv Ezf. r] BE aQa TCQbg t^v E^d (iSL^ova 
X6yov s%SL VjicsQ 6 EHS to[isi)g TCQbg tbv EZ0 toiiia. 
d)g Sl 6 to fisijg TCQbg tbv tofiia^ ovrog fj i)7cb HE® 
ycovCa iCQbg tijv i)7cb ZES ycjvCav. i^ BE aQa 

4. rd\ corr. ex BT BVat., BT v. 6. 
7. TtQOGTtLTttsra) Bv et Vat., sed corr. 



1. tj'] -a-' codd. 
ag] om. VBVat.mv. 



EUCLIDIS OPTICA. 15 

Aequales et aeqnidistantes magmtndines inaequa- 
liter distantes ab oculo non proportionaliter spatiis 
uidentur. 

sint duae magnitudines ah et gd inaequaliter di- 
stantes ab oculo e. dico, quod non est, sicut apparet 5 
habens^ gd a,d ah^ ita he ad ed. accidant enim duo 
radii ae^ eg^ et centro quidem e, spatio uero ez de- 
scribatur periferia izt quoniam ergo ezg trigonus ezi 
sectore maior est, ezd uero trigonus ezt sectore minor 
est, trigonus ergo ezg ad ezi sectorem maiorem pro- lo 
portionem habet quam eed trigonus ad ezt sectorem. 
et permutatim ezg trigonus ad ezd trigonum maiorem 
proportionem habet quam ezi sector ad ezt sectorem, 
et componenti egd trigonus ad ezd trigonum maiorem 
proportionem habet quam eit sector ad ezt sectorem. 16 
sed sicut egd trigonus ad ezd trigonum, ita recta gd 
ad rectam zd. at uero gd rectae ah est aequalis, et 
sicut ah ad dz, ita he ad de. et he ergo ad ed 
maiorem proportionem habet quam eit sector ad ezt 
sectorem, sicut autem sector ad sectorem, ita iet 20 
angulus ad zet angulum. recta ergo he ad eei rectam 



9. Ante sectore {pr.) del. ad D. 10. Post est del. est 

ergo D. 12. ezg] corr. ex ezd D. 

9. AE, ET] mut. in EB, EA m. 2 Vat. 11. rw] rd v. 

12. TflS] r6 V. 13. yivyiXov] ov Bv; ov Vat., corr. m. 2. 15. 
(t,slt(ov V. iGtl Vat.mv. 16. iGtl Vat.mv. 19. rd(alt.)] 
t6v V. 20. Ante 6 ras. 1 litt. Vat. EZH] EZ v. tov] 
triv V. 21. tQvyaivco v. TCQog — 22. tqiyoivov] bis v. 22. 
to\iBvg\ toiisG B. 23. EJT] EFJ m. 24. JZ] dlHl V, 

item lin. 25. 28. HE@] in ras. V. 29. ZE0] in ras. V. 

ycDviav] ycovicc v. BE] corr. ex BE^ m. 2 V, om. Bv, 

add. m. 2 Vat. &qcc] m. 2 Vat., om. Bv; evd^stu add. m. et 
m. 2 Vat. 




16 EUCLIDIS OPTICA. 

TCQog tiiv Ejd ^SL^ova k6yov i%si ^tcbq fi ijco HES 
yaovCa Ttgbg t^v 'b%h ZES, xal ix ^lv tfig ijtb HE® 

, ycovLug ^kijCBtai tb Fz/, bk 8\ tf^g 'bicb ZES tb AB. 
o^x &v&koyov aga totg aTCo^tiliia^LV bQatai to: t6a 

5 ^ByBd"r]. 

To: dQd^oychvLa fiByBd^rj i^ aTCO^f^fiatog 6Q(hiiBva 

xsQLq)BQrj q)aLVBtaL. 

i6tca yaQ dQd^oytovLOv tb BF 
10 B6thg ^Btia)QOv i^ aTCo^tilfiatog 

6Q(hfiBvov. ovxovv^ iicsl Bxa6tov 

t&v 6Q(o^iv(ov i%BL tL ^flxog aico- 

^tifiiiatog^ 01) ysvdfiBvov oi>KitL 

bQataL^ 71 iiBV r &Qa ycovCa oi)^ 
16 ^QataL,^ tic di ^^ Z 6rj^Bta ^6vov (paCvstaL, 6(ioC(Dg 

xal i(p' Bxd^trjg t&v Xoltc&v ycnvL&v tovto 6v^^ifi6Btat. 

&6tB oXov 7tBQL(pBQhg (pavif^6BtaL. 

L . 

T&v 7i&t(o tov Sftftarog xBLfiivcav iTCLTcidcov tcc tcoqqcd 
20 (iBtBOQdtBQa (paCvBtaL. 

B6ta) (i^^a tb A ^BtBCJQdtSQOv xsCfiBvov tov BEFy 
Tcal 7CQ067CL7Ctitcj6av dxttvBg a[ AB^ AE^ AA^ AF^ 
i)v rj AB xdd^Btog B6tco i7cl ro {)7toxBCiiBvov i^cC^csSov. 
XiyG)^ otL tb FA tov AE fiBtscoQdtBQOv (paCvBtaL^ tb 
26 d^ AE tov BE. slXifi^pd^G) ydQ iTcl tfig BE tv%bv 
6i^fiBtov xatd ro Z, xal i^^d^co 7CQbg ^Qd^dg i^ ZH. 
[xaY] i^csl at SijJBLg 7CQ6tBQOv 7CQbg ti^v ZH 7Cqo6- 

7CC7CtOV6LV fl7CBQ 7tQbg tr^V Zr^ 7CQ067CL7CtitG} tfl ZH ^l 

1. TCQhg xi]v Ed] yavla^ corr. in svd^stcc Jtgbg xr\v EJ 
m. 1 V. Post EJ add. svd^stav Bv, svd^sta Vat. (v m. 2). 



EUCLIDIS OPTICA. 17 

maior proportio quam iet angulas ad 0et angulum. 
et ex angulo quidem iet maior gdy ex angulo uero 
J3et recta minor ah. non ergo distantiis proportiona- 
liter uidentur magnitudines aequales. 

Recta^gulae magnitudines e distaaitia uisa* peri- 5 
feriae apparent. 

esto enim rectangulum hg existens eleuatum e 
distantia uisum. igitur quoniam unumquodque uiso- 
rum habet longitudinem distantiae, qua facta non iam 
uidetur, angulus g quidem non uidetur, puncta uero 10 
d, z tantum apparent. similiter et in unoquoque reK- 
quorum angulorum hoc continget. quare totum peri- 
fer[ia] apparebit. 

Sub oculo iacentium planorum remotiora quidem 
eleuatiora apparent. 15 

esto oculus a eleuatior iacens quam hedg^ et ac- 
cidant radii ah, ae, ad, ag, quorum ah recta cathetus 
esto super subiacens planum. dico, quod gd quam de 
eleuatius apparet, sed et de quam he. sumatur mhe 
punctum z, et trahatur perpendicularis zi. quoniam 20 
uisas primum accidunt ad zi quam ad zg, accidat ei; 



12. perifer sec[. ras. D. 21. primum] scr. prius. 



2. ^fv] ^o' BVat. 3. ^XBnBxciL] (istSov VBVat.mv. 6. 
<&•'] t' codd. 7. &7to6rrnicitG}v v, comp. BVat. 9. dp^O"©- 

ydoviov V. 10. iato)^ V, iatSc BVat.mv. 13. ysvoiiivov 

VBv, yivoiisvov Vat. 14. F] ydg (per comp.) F Y. 16. 

nai'] nal 7] BVat.v. ixaatov m, k^datriv V. 17. q^avrjastai] 
avti§T/iastaL Vm. 18. t'] ta' codd. 21. BEF] BEN By. 

22. rcQoaitirctstco Vat.v. 25. tfig] t6 v, tov Vat.B. 26. 
'Katd] om. BVat.v. 27. tial] supra scr. m. 1 V, om. BVat.v. 

al] corr. ex ovv Vat. 28. TrQOCTtiTtTEToacsav ^^V..^ ^^^ ^'ot.. 

Euclides, edd. Heiberg et Menge. YLI. ^ 



18 



EUCLIDIS OPTICA. 




^lv Ar xard: ro H 6riiietoVj 'fj Ss AA xata ro ©, ij 

S\ AE xatci ro K. ixsl ovv tb H tov ® i6ti ^steca- 

gdtSQOVj tb Ss 

tov K^ aAA' iv S 

6 s6ti tb H, iv rovro) 

ro r, iv S Sh tb ®^ 

iv tovt(p tb A^ iv 

c5 8\ tb jSl, iv tovto 

tb Ej Sict Ss t&v 

10 Ar^ AA i\ JT (paLvstaL^ Sl& Si t&v AJ^ AE ^ ^E^ 

fl FA aQa trjg AE [istsoQOtiQa q)aCvstaL, b^ioCcoQ xai 

il z/JE tYis BE ^stscoQOtSQa cpavT^^staL' ta yo^Q ixb 

listscjQOtSQmv axtCvcjv bQch^sva ^stscoQotSQa q^aCvstai, 

xal (pavsQdv^ 8tL ta iv iistS(OQG} xsC^isva xotXa 

15 (pavri6staL, 

La\ 

Tcjv av(D tov biiiiatog xsLfiivcav iitLJtsScav tcc 7t6QQ(o 

taTCSLvdtSQa (paCvstaL. 

i6tc3 (ififia ro A taitSLvdtSQOv xsC^isvov tov BF 
20 iTCLTtsSov^ xal 7tQ067tL7ttstc36av dxttvsg aC BA^ AA^ 

AE^ AF^ S)v Yi AB xdd^s- 

tog i0t(X) iitl tb {)7toxsC(isvov 

i7tC7tsSov. Xsy(o^ otL tb FE 

tov EA ta7tSLv6tSQOv (paC- 
25 vstaL. Slcc di) ro ^tQOSxtsd-sv 

d^ed^Qrjiia ta7tSLvotiQa i^ (ihv 

AF dxtlg trig AE^ ^ Sh AE trjg AA^ fi Sh AA 
^ tfig AB. dXXd Sl^ ^lv tmv FA^ AE tb TE ^XsTtstaL^ 

SlA Sh t&v EAj AA tb EA^ SlA Sh t&v AA^ AB 

6. toijto V. 10. AF, AJ] corr. ex AF (F in ras.) V, 

FJBy, et Vat., sed corr. m. 2; supra scr. ScKtivtov m. 2 Vat. 




EUCLIDIS OPTICA. 19 

quae est zi^ recta ag ad punctuin i et ad ad punctum f, 
sed ae ad punctum Jc. quoniam ergo i punctas quam t 
eleuatior est, t uero quam ifc, in qua uero est i, in 
ea est g^ et in qua f, in ea d^ in quo h^ in eo e, per gd 
uero ea, quae est gd^ apparet, per ed autem ea, quae 6 
est de, ergo gd quam de eleuatius apparet. similiter 
autem et de quam he eleuatius apparebit. sub ele- 
uatioribus uero angulis radiis uisa eleuatiora appare- 
bunt. 

et manifestum est, quod in eleuato iacentia con- 10 
caua apparebunt. 

Super oculum iacentium ebipedorum remotiora qui- 
dem humiliora apparent. 

esto oculus a humilior iacens Ig ebipedo, et ac- 
cidant radii ha^ ad, ae, ag, quorum recta ah chathetus 15 
esto super suppositum epipedum. dico, quod ge quam 
ed humilior apparet, ed uero quam dh, per prae- 
missum 3. theorema ag quidem radius humilior est 
quam ae et ae quam ad et ad quam ah, sed per ga 
et ae ge uidetur. sed per ea et ad ed, per da uero 20 



8. angulis] deUndum. 16. epipedum] corr. ex epei- 

pedum D. 



jr] JN Y; corr. ex ^T m. 2 Vat. Ante Sid magna ras. 
Vat. Twr]8upra scr. v. AJ^AE] EJ VBVat. v, corr. m. 1 V, 
corr. m. 2 Vat. 12. BE] B in ras. v. 13. 6q[l6)[lsvcc B, et 

Vat., sed corr. m. 2; 6q6)- in ras. v. 14. or m. iiststoQo- 
rsQot Vat.m. 16. ta'] i§' codd. 20. inmsSoi v. itQoa- 
niiixsxai Bv. 25. Ante 8id add. xb 8\ EJ rov' JB BVat.v. 

26. xoc%siv6tSQOv V. 28. AB] inter A %t B ras. 1 litt. v. 

AE] tb AE V. 29. EA] AE v. AK\ Azl ^. 



20 



EUCLIDI8 OPTICA. 



rb jdB (paCvBtaL. tb FE &Qa rov E/l taTCSLvdrsQOV 
q^aivsruL^ rb Si Ejd rov jdB. 

T&v slg roijiiTtQO^d^ev ^rixog i%6vrG}v ra ^ihv iv rotg 
5 de^Lotg elg ra &QL6rsQa doxst xaQrlxd^aL^ ra ds iv rotg 

aQL6rsQ0tg sig rcc 8si,Ld. 

s6ra) di5o bQcofisva iisyid^ ra AB^ P^, oftfto; 8% 

i6ra) ro E^ dq)^ ov 7CQ067tL7trira)6av dxrtvsg at E®j 

EKj EA^ EZ^ EH, ER Xiyw^ 
10 Srt a[ ^lv EZ^ EH^ ET do- 

7COV6LV slg rd dQL6rsQd iisrfjxd^aL^ 

at 81 Ee, EK, EA slg rd 

8s^Ld. iitsl ydQ ij EZ rfig EH 

i6rL Ss^LOiriQa^ fi 8h EH rfig 
15 Er^ ivrsvd^sv aQa ff EF rfig 

EH 8oxst slg rd dQL6rsQd iier- 

fjX^aL^ il 8h HE rf^g EZ. 

6fiOL(og xal at EK^ EA^ ES 8oxov6lv slg rd 8si,Ld 

fisrfjx^f^i" 
20 Ly\ 

T&v t6(QV fisysd^&v xal vTtb ro airb S^fia xsLfii- 
V(DV rd jtdQQCD ^srs(0Q6rsQa (paCvsraL. 

i6ra) t6a ^syidn] rd AB^ FA^ EZ^ H^^a 8h i6ra) 
ro H fisrsG)Q6rsQ0v xsC^svov r&v ^syed^&v^ xal 7tQ06- 
25 jtL7trirci)6av dxrtvsg at HA^ HF^ HE. Xiyo)^ orL ro 
AB rov FA (isrsc3Q6rsQOv (paCvsraL, rb 81 FA rov EZ. 
iTtsl ydQ fi HA rfjg HF i6rL fisrsojQoriQaj ij 8h HF 
tfig HEj xal iv 6 s16lv at HA^ HF^ HE^ sv rovro) 




2. z/B] JE Vm. 3. ifl'] ty' codd. 
Bv. 7—8. ^ffrco $h 6>^a BVat.v. 8. 



4. rb fyjtQocQ^sv 
dyitlvss] e corr. Vat. 



EUCLIDIS OPTICA. 21 

et ah dh apparet. ergo ge quam ed apparet et ed' 
quam dh liumilior. 

In ante habentium longitudinem quae quidem in 
dextris, in sinistra, quae uero in sinistris, in dextra 
educi uidentur. 5 

sint duae conspectae magnitudines ah et dg, ocu- 
lus 6, a quo accidant radii et et eh^ ea, ez, ei, eg. 
dico, quod ez et ei ei eg uidentur in sinistra pro- 
tractae, et uero et eh et ea in dextra. quoniam enim 
60 quam ei dexterior est, ei uero quam eg, inde ergo 10 
et ab ei uidetur in sinistra tracta, ei uero ab ez. 
similiter eJc^ ea, et uidentur in dextra tractae. 

Aequalium magnitudinum et sub eodem oculo 
iacentium longius iacentia eleuatiora apparent. 

sint aequales magnitudines 15 
ah^ gd, ez, oculus uero sit i 
eleuatior iacens magnitudinibus, 
et accidant radii ia et ig et ie. 
dico, quod ah quam gd eleuatius 
apparet, gd uero quam ez. quo- 20 
niam ergo ia quam ig est ele- 
uatior, ig uero quam ie, et in quibus sunt ai et ig 

11. ef] scr. eg. uidetur] corr. ex uidentur D. 18. 

ia] a D. 

9. Er'\ EN Y (inB v et y difficulter dignoscuntur). 10. 
EF] EN V. 14. iGTl V. 15. Er(alt.)] EN v. 16. EH] 
OH V. 17. HE] EH m. 18. al] om. Vv. Sd^ovav 

BVat.vm. 20. ty'] td' codd. 22. Post TtOQQoa add. xa/- 

^LSvcc Vat.v, et supra scr. B. 23. ^fta v. Ss] bis Vat., 

sed corr. 24. ngoGTCt^tiro} Bv. 27. HA] H e corr. B. 

ietiv V. 28. fl)] mut. in ols V, olg BVat.vm. ro-uxcal^ 

xo^dtotg VVat.m, obscuro comp. B, TovToav ^. 




22 



EUCLIDIS OPTICA. 



i6tl xal %k A^ F, E 6rj^sta^ iv S Sh ric A^ jT, E, iv 
tovtp xal t& ABj r^j EZ ^syid^rj^ tb AB aga tov 
FA fiets(OQ6t£Qov (paCvstai^ ro d\ FA tov EZ. 

5 T&v l'6(ov ^sysd^&v xal avotSQO) tov o^i^atog xsv- 

^sviDV ta 7t6QQ(0 ta7CSLv6tSQa (paCvstav. 

s6t(o t6a fisyid"rj tk AB^ jTz/, EZ ^stS(X)Q6tSQa 

xsCiisva tov biifiatog tov H. liyco^ ott, tb AB tov F^ 

tcatSLv6tSQ0v (paCvstaL^ tb ds FA 
10 tov EZ. 7tQ067CL7ttit(o6av dxtLVsg 

at HB, Hz/, HZ. i^tsl olv fi HB 

dxtlg tflg HA i6tL ta^tSLvoti^a^ 

^ d^ HJ tfig HZ^ dXV iv S s16lv 

ai HBj HAj HZ^ iv tovtto i6tl 
15 xal td B^ jd^ Z ^riiista^ iv S dh 

td B^ Aj Z^ iv tovt(p xal td 

AB^ r^, EZ iisyid"^^ tb ^isv AB 

&Qa rot) FA ta7tSLv6tSQOv (paCvstaLj tb dl FA tov 

EZ \ta7tsLv6tSQ6v i6tLv\ 




20 



LS 



'^O^a dXMiXoDV v7tSQi%SL V7tb ro avtb ()[i^a xsC^sva^ 
7tQo6L6vtog iilv tov biifiatog ^sCtpvv ^st^ov tb v7tSQ- 
(paLv6(isvov (paCvstai^ d7tL6vxog 8\ iXd66ovL. 

i6t(D d^io avL6a [isyid"rj td AB^ F^, ^st^ov dh i6t(o 

25 ro AB^ '6ii[ia Sh i6t(o tb JS, d(p^ ov 7tQo67tL7ttit(o dxtlg 

dLd rov r r^ EZ. i^tsl oiv V7tb tov &^[iatog xal tfig 

EZ dxttvog td ZB^ F^ (paCvstaL^ tb AB &Qa tov Fjd 



1. iaxl'] sIgI m. I^Cpr.)] ^ v. co] olg m, corr. ex 

^ V. r, E] e corr. V. 2. rovrotg m'. 3. FJ (alt.)] 



EUCLEDIS OPTICA. 23 

et ie, in eis siint et a, g puncta, in quo nero a, g, e, 
in eo et db, gd, ez magnitudine; igitur ab quam gd 
eleuatior apparet et gd quam ez, 

AequaKum magnitudinum atque superius oculo 
iacentium remotiora quidem humiliora apparent. 6 

sint aequales magnitudines dh, gd, ez eleuatiora 
iacentia oculo i, dico, quod ah quam gd humilius 
apparet, gd uero quam ez, accidant enim radii ih, id,iz. 
quoniam ergo ih radius id humilior, id uero quam iz, 
sed in quo sunt ih, id, iz, et in eo sunt et h, d, z 10 
et ah, gdy ez magnitudines, ah ergo quam gd humilior 
apparet et gd quam ez, 

Quaecimque altemorum se superant sub eodem 
oculo iacentia, accedente quidem oculo maiori maius 

j^ superapparens apparet, ab- 15 
cedente uero minus. 

sint duae inaequales mag- 
nitudines ah, gd, maior- 
que sit ah, oculus autem 
sit e, a quo accidat radius 20 
per g ez. quoniam ergo 
^ sub oculo et ez radio zh 

et gd apparent, ah ergo ei, quod est gd, super- 

2. magnitudine] corr. ex magnitudo D. 

^P m. 4. l8'^ lb' codd. 6. toLTCBivoiXBQa v. 10. itQoa- 
TCLTcrha) Bv. 11. iytsl ovv — 13. HZ] bis, sed expunctum V. 

15. J] z/, E v; J, H e corr. B. 16. To^drotg VBVat.v. 

18. ta^8iv6t£Qa v. 19. taTtSLvdtsgdv iativ] om. BVat. v, taTCBL- 
votSQov m. 2 Vat. Post iatcv add. r^ k^fjg V, m. 2 Vat. 

20. wH tg' codd. 22. ftetfort (isitov'} -tovi (isi- postea 

additum V, (islSovl mg. m. 2 Vat. v^ocpaivdiisvov Vat., corr. 
m. 2. 23. ^Xaaaov BVat.v, corr. m. ^ \«A». 




24 



EUCLIDIS OPTICA. 



(ilifia iyyvTBQca xal i6r(o rh H, «9?' 0'5 7CQ06itncrircj 
axrlg dict rov F i^ H®. iTtel ovv iTtb rov ^[i^arog 
xal rfjg H® dxrtvog q^aCverai rb F^ xal rb @B^ rb 
^5 AB aga rov Fjd iiet^ov q)avi^6eraL ra A0, ifiXeTCsro 
8i i)7cb roi) E rm AZ ^iet^ov^ ^et^ov d% rb AS rov 
AZ. 7tQo6L6vrog ^ev aQa rov Hfiiiarog ^iet^ov ro i)7CeQ- 
q)aLv6iievov q^aCveraL ^eC^ovL^ aiCL^vrog 8e iXarrovt 
[(paCveraL rb {)JceQq)aLv6^evov fiet^ov^ 



10 



L% . 



X)6a «AAiJAoi/ {)7CeQixeL ijcdvm rov S^fiarog &vL6a 

^eyi%"Yi^ 7CQ06L6vrog ^ilv rov &iifiarog iXd66ovL ^iet^ov 

(paCveraL ro 'b7CeQ(paLv6(ievov^ d^CL^vrog 81 [ibC^ovl. 
i6r(o aVL6a ^eyid^rj rd AB^ FAj hv ^et^ov rb AB. 
15 i6ra) Sftfto; rb E^ d(p' o-S jCQO^TCLTCrirco dxrlg 8Ld rov F 

fl EZ, iicel ovv vicb 

rrjg EZ dxrtvog dico- 

ka^^dveraL rd ZB^ 

FA ^eyid^rj^ rd BZ^ 
20 FA aQa l'6a dkXriXoLg 

(paCveraL. rb AB ccQa 

rov FA ^iet^ov (paC- 

veraL r& AZ ^eyid^eL. 

TCQO^i^Xd-cj 8ii rb o^ifia iyyvriQco xal i^rco ro H^ 
26 d^p* oh 7CQ06JCL7crircD dxrlg S^d rov F fi H®. i^cel 

ovv 'b^cb rrlg H@ dxrtvog d^coka^^dverai rd BG^ 

r^, {)7cb 8h rrig EZ rd ZB^ FA, i6rL 8h ro ZA 

roi) A® ^et^ov^ 7CQ06L6vrog ^hv ccQa rov ^iiiiarog 




vM' 



1. fisysd"ri V. 3. HyMtog v, ut saepe. 4. rb FJ xa^] 

mg. m. 2 V, om. Bv, m. 2 Vat. &B] B in ras. V. 



EUCLIDIS OPTICA. 25 

appaxet az magnitudine. transmoueatur oculus propius 
et sit iy a quo accidat radius it per g, quoniam ergo 
sub oculo Qt it radio apparet thj ergo db eo, quod 
est gdy maius apparebit eo, quod est at uisum est 
autem sub ez az, maius autem at quam az. itaque 5 
accedente quidem oculo maiori maius apparet super- 
apparens, abscedente uero minus. 

Quaecunque alternorum se superant super oculum 
inaequales magnitudines, accedente quidem oculo mi- 
nori minus apparet superapparens, abscedente uero 10 
maius. 

sint quidem inaequales quantitates ah, gd, quarum 
maior ah^ et oculus e, a quo accidat radius ez per g, 
quoniam ergo sub ez radio continetur zh et gd magni- 
tudo, ah ergo quantitas quantitate gd maior apparet 15 
eo, quod est az. attrahatur autem oculus prius et 
sit i, a quo accidat radius it per g, quoniam ergo 
sub it radio deprehenditur ht et gd, sub ez uero zh 
et gd, est autem za quantitas quantitate at maior, 



5. ez — af] in ras. m. 1 D. 16. prius] ser. propius. 



6. ipXsTtstG) V. 6. irco] r6 BVat.v. iisttov] om. VBVat.mv. 
Ss (alt.)] om. m. 7. rov diiiLatog] supra scr. m. 2 B. (isiSovi 
lisTtov B. 8. iisiSovi] om. Bv, m. 2 Vat. ^Xattov BVat.v, 
corr. m. 2 Vat. 9. q^aivstai — {Lsttov] om. Bmv, m. 2 Vat. 

10. ts'] li' codd. 11. iTtdvfo] supra scr. V. 12. iXaGCovC] 
supra scr. m. 2 V. 13. tb vnsgq^aivdiisvov (paivst ai m. ftf/- 
tovi] in ras. V. 14. t6 — 16. ^ftfta] forG) tb AB, oiiiia di 

Vat.^m. 14. AB (alt.)] AB %ai Vat.v. 16. ^116] 'bitSQ v. 

18. ZB] in ras. V. 19. ra BZ, Tz/ aQa] om. m. 23. 

rw] t6 V. {Lsys^-ri v. 24. dif\ §8 Va\,.^. Yl. Wv-v ^. 



26 EUCLEDIS OPTICiL 

ikd66ovL fiBt^ov to i7t6Qq)aLv6iievov (paivetai^ &Mivtog 

ds ILBltfiVl \}LB1^0V'\. 

T9<^a &klriX(ov vmQBXBv^ ijt' sid^Biag tp ikdttovi 

5 fiByid^BL tov o^fiatog 7CQo6L6vtog tB xal a<pL6taiidvov 

t& t6G} &bI 86i,BL tb {)7tBQg)aLv6fiBvov tov ikdttovog 

V7CBQi%BLV. 

i6t(D dvo &vL6a iLByi%^ri ta AB^ Fz/, ©v {LBttpv th 
AB^ '6^[ia di i6t(o tb Z iit^ Bvd^BLag TCBifiBvov tm Tci- 
QatL tov Fjd ^Byid^ovg t^ F. ^ 
kiyco^ otL tov Z (i^^atog 
7CQo6L6vtog xal d<pL6tafiivov 
ijr' Bi^d^BLag 6vtog t& t6(p 
86%BL 'b7CBQq)aLVB6%'aL tb AB 

5 tOV r^. 7CQ067tL7Ctit(D yoCQ 

dxtlg SLa tov F f} ZE. tb 

AB &Qa tov r^ i^CBQ^paCvBtaL tp AE. ^BtaxBXLVij^d^cs 
dii tb ^(i(ia xal iatco a7t(otiQ(o xal i6t(o i%^ Bid^BLag 
tb H. 'fj &Qa &7tb tov H S^fiatog axtlg 7tQo67tL7ttov6a 
50 ikBv6BtaL dLa roi) F 6rjiiBL0v xal 7tQ06BVB%%"i^6BtaL (ii%QL 
tov E 6r}^BL0v^ xal rp aifta 'b7tBQ(pav7J6BtaL tb AB 
tov FA. 

Ly{ . 

Tb Sod^BV vtl)og yv&vaL^ 7trjkLK0v i^tCv^ ijkCov (paC- 
J5 vovtog. 

i'6t(o tb Sod^hv iitl)og tb AB^ xal Siov ax)tb yv&vcUj 
7t7jkCzov i6tCv. i6tco [ihv ^(i(ia tb A^ ijkCov Sh axtlg 



E r z J^ 



2. fifirjov] om. Vat.^mv, m. 2 Vat. Dein add. ~l|^s V, 
m. 2 Vat. 3. tf'] ni' codd. 6. tc5 tcqi icBi'] in ras. m. 1 v. 



EUCLIDIS OPTICA. 27 

accedente ergo minori miims superapparens apparet, 
abscedente uero maius. 

Quaecunque altemorum se superant, in directo 
minori quantitati oculo accedente et abstante aequali 
semper uidebitur superapparens minorem excedere. 6 

sint duae inaequales magnitudines ah et gdy qua- 
rum ah maior, oculus uero sit in directo iacens 
termino quantitatis gd ei, qui est g, dico, quod 
puncto g oculo accedente et abstante in directo exi- 
stente aequali uidebitur superapparens ah ei, quod 10 
est gd. accidat enim radius ie per g, itaque ah ei, 
quod est gd^ superapparebit eo, quod est ae. trans- 
moueatur autem oculus et sit longius et sit in directo 
ei, quod est i. ab oculo ergo radius accidens ueniet 
per g punctum et adiungetur usque e punctum, et 16 
eodem superabit ah quidem gd. 

Datam altitudinem cognoscere, quanta sit, sole 
apparente. 

esto data altitudo afe, proponaturque eam cognoscere, 
quanta sit. sit oculus d^ solis autem radius ga con- 20 



1. minori minus] in ras. D. 2. maius] in ras. D. 3. 
se] mpra scr. D. 11. accidit D. ie\ scr. ze. 13. 

directe D. 



iXccGGovog Vat.v, comp. B. 9. si^&siag'] -ag in ras. v. 

r^] rov m. 11. tov Z] ro Z m. 1 V, tA T rov B Vat.Vat.^mv, 
m. 2 V. 16. ydg] om. m. 16. ZE] EZ m. 18. tb ^ft^al 
tov ^iniatog v. &7totSQ<o VBv, et Vat., corr. m. 2. httD] 
TisiGd^to m. 19. t6]t^ m. H (alt.)] om. Bv, m. 2 Vat. 

20. did] %ccl Sloc B Vat.v. 23. tT]'] l&' codd. 24. d^xC^ 
i6tl tov V. 27. iati v. 



28 



EUCLIDIS OPTICA. 




7} FA 6vii^aXXov6a rp jteQatL rov AB (leydd^wg xal 

Sl7Ix^(o ^sxqI' tov A (iii^arog. i6t(o dh 6oci& ^ AB 

tov AB. xal KsC^^^m 

etSQdv ti ^dysd^og tb EZ 
5 6v^pdXkov tri axttvL fiii 

Ttdvtojg Tcatavya^d^svov 

{)7t^ ai>tijg xatic tb Z Ttd- 

Qag. ^Q(io6taL oi)V slg 

tb ABA tQ^yovov sts- 
10 q6v tL tQLycDVOv tb EZA. i6tLV aQa^ hg fi AE 

TtQbg triv ZE^ ovtoog fi AB %Qbg ti^v BA, dlX 
- 6 tr[g AE TtQbg tijv EZ X6yog s6tl yvciQL^og' xal 

6 tfjg AB aQa TtQbg tijv BA l6yog i6tl yvd)QLiiog. 

yvd)QL[iov dh tb AB. yvd^QL^iov aQa xal tb AB. 

16 , Ld'\ 

Mii iyjtaQXOvtog fjkLOv tb Sod^hv vtl^og yvovaL^ Ttri- 
kCxov i6tCv. 

iatm tL [jisysd^ovg^ vifjog tb AB^ S^fia Sh i6t(o 

tb -T, xal Siov i6t(o tb AB yv&vaL^ jti^kCxov i6tCvy 

20 &g ^ij ijtaQxovtog fiXCov. xsC^d-o xdtOTttQOv tb ^Z, 

xal nQ06sx^s^kif[6%^(o tfj EA iit^ sv^sCag i\ z/J5, ^X9^ 

oh 6v\i^aXsl t(p TtsQatL tov AB ^sysd^ovg t^ JJ, Tcal 

7tQ067tL7ttit(o dxtlg &7tb tov &fi(iatog tov F ij JTif, 

\ xal dvtavaxsxXd^d^cj j axQLg o-S ^vfi^akst rra Tti^atL 

25 rov AB [isyid^ovg to A^ xal 7tQo6sx^s^lifi6%^(o ty zJE 

ri E&^ xal ^x^^ ^^^ '^^^ ^ ^^^ '^^'^ ^^ xdd^stog ij 

7. Ante xara add. ScXXd m, m. 2 VVat. 8. i]Qii66&oi} m, 

9. ABJ] corr. ex ABF Y. 10. z/E] z/Z Bv, et Vat., corr. 

m. 2. 12. JE] JZ Bv, et Vat., corr. m. 2. EZ] in ras. V, 

ZE BVat.v. 14. Post AB add. :~^|^5 V, m. 2 Vat. 16. 

t-O"'] x' codd. 17. i6ti v. Dein add. k^fjg B, sed deL 



EUCLIDIS OPTICA. 29 

cidens termino a magnitudinis et protrahatur usque 
ad oculum. sit autem umbra db altitudinis ab, iaceat- 
que altera quantitas ez concidens radio non omnino 
illuminata ab eo secundum z terminum. aptatus est 
ergo ut ahd trigono alter trigonus ezd. est ergo sicut 5 
de ad ze, ita dh ad ha, sed de ad ez proportio est 
nota. et dh ergo ad 6a proportio est nota. notum 
autem dh. ergo et ah, 

Non existente sole datam altitudinem, quanta sit, 
cognoscere. lO 

esto altitudo afe, oculus uero sit g, et sit proposi- 
tum ah cognoscere, quanta sit, sole non existente. 

iaceat speculum dz, et 
adiciatur rectae ed m d 
puncto dh, terminus 15 
eius coniungatur ter- 
mino quantitatis ah, qui 
est 6, et accidat radius 
ab oculo g gi, et re- 
fringatur terminus eius et coniungatur termino a a6 20 
magnitudinis, et adiciatur rectae de recta et. tra- 




5. ut] scr. in. 6. de {utrumqtiey] e in ras. D. 13. Supra 
speculuin add. planum D. 19. oculio D, sed corr. 20. a] 
corr. ex ah D. 21. recta] corr. ex recte D. 



18. (liysd^og m. 19. iari v. 20. atg] om. Bv, m. 2 Vat. 

rov rjUov m. 22. av(ipaX^ V, sed corr.; Gv^pdXXjj BVat.v. 

Post B del. xai TtQoasTi^B^Xriad^o) tjj JE ij E0 B. ' 24. &va- 
liSTtXdad^Gi B et Vat., sed corr. ; SivaTiSTiXilG&G} v. av^paXfj V, 
sed corr.; av^^dXXrj BVat.v. 25. AB] corr. ex ^B Y. *L^. 
^ (pr.)] supra scr. Y . 



30 EUCLIDIS OPTICA. 

rS. ijtel oiv 7tQ067te7tra)xsv dxtlg fj JTH xal avtavu- 
xsxka6rav i^ HA^ TtQog t6ag yaovCag &vaxs7cXa6(idv(u 
sl6LV^ i)g iv rotg KaroTtrQLXOtg Xiysrar £67} &Qa ymvCa 
il i)7tb FH® rfj iTtb AHB. akka xal ii ixb ABH 
5 T^ 'bjtb r®H t6rj' xal XoLTtij aQa '^ imb Hr@ loixfj 
ry i)7tb HAB i6rvv t6i^. l6oy(ovtov &Qa i6rl rb AHB 
rQLycDVOv rc5 FHS rQiyfhvfp. r&v S\ l6oycavca}v rQi^- 
ymvcDV avdkoydv si6iv at %ksvQaC. i6rvv aQa^ &g ij 
rS TtQbg rijv QH^ ovrfog i^ AB TtQbg rijv BH aXX' 
10 6 rfig r& TtQbg rrjv &H X6yog i6rl yvdiQt^og' ocal 6 
rrjg BA ccQa TtQbg rijv BH Xdyog i6rl yvd)Qi^og. &XX^ 
'^ HB i6rL yvd)Qifiog. xal ii AB aQa i6rl yvdiQLfwg. 



X . 



Tb dod^sv pdd^og yv&vat^ Ttr^XCxov i6rCv. 
15 i6rci) rb 8o%^hv ^dd^og rb AA^ 'dfifia Sh i6ra} rb E^ 

xal Siov rb pdd^og yv&vat^ Ttr^kCxov i6rCv. 7tQ06XL7trirG) 

ydQ rfj ^tl^si iikCov axrlg ri 

EA 6v(ipdkkov6a r^ ijtt- 

7tiS(p 3car«; ro B 6rjiistov 
20 Tcal rp pdd^SL xard rb A. 

xal TtQO^sxPspkTJ^d^co djtb 

rov B iit^ sid^sCag i} BZ^ 

xal i^xd^ci) ajtb rov E iitl 

rijv BZ sid^stav xdd^srog 
25 ii EZ. i%sl ox)v t6r} ycjvCa ij iitb EZB rfi iitb BAA^ 

&XXd xal ii {)7tb ABA rfi V7tb EBZ^ xal ii rQCrrj &Qa 

il i)7tb BEZ rfi i)7tb AAB i^riv t6r}. i6oyd)vcov &Qa 

i6rl ro AAB rQCycavov rc5 BEZ rQiydivcp. xal at 

1. 6Lvav.iyLXa6xoci Bv et Vat., sed corr. ; 6LVXccvociiByLkaTm m. 
4. rH&\ in ras. m. 5. Xoini^ Xoi7t6v By. Xoltc^} Xomoi v, 




EUCLIDIS OPTICA. 31 

hatur ab ocnlo g snper et cathetus gt quoniam 
ergo accidit radius gi et refriiigitur ia^ ad aequales 
angulos repercussi erunt. aequalis igitur angulus t 
angulo i, et reliquus ergo reliquo. aequiangulus ergo 
tgi trigonus aih trigono. est ergo sicut gt ad ti, 6 
ita et ah ad ii, sed quantitatis gt ad ti proportio 
est nota. et 6a ergo ad 6i proportio est nota. sed 
hi nota. ergo et 6a est nota. 

Datam profunditatem, quanta est, inuenire. 

esto data profunditas ad^ oculus autem sit e^ sitque 10 
propositum cognoscere, quanta sit. accidat autem 
radius ed concidens plano ad punctum h et profundi- 
tati ad punctum d, et adiciatur a puncto 6 in directo 
6 1)0^ et trahatur ab e super hz cathetus ez. quoniam 
ergo z Qi a anguli sunt aequales, et 6 contra se positi, 16 
erit et tertius tertio aequaKs. quare trigoni similes. 
latera igitur proportionalia. est igitur sicut ez ad zh. 



2. aequales] eaquales B. 4. t] eras. D. 6. ti] 

t B. 



Xoi B. 6. HAB'\ '^a/S m. ird] xov v. 8. &Qa\ supra scr. B. 

9. ^B] ^H Bv et Vat., sed corr. 10. r&\ FO Bv. yvmQi- 
itdg icti BVat.v. 13. x'] xa' codd. 14. i6xi v. 16. 

xb JE, xal 8iov\ om. Vat., tb E ins. ante ^(rro, xal 8eov post 
forcB m. 2. 16. Siov hta) Bv. ieti Vat.vm. 17. r^ 

&il)£i r}Xiov\ om. v, m. 2 Vat. 18. Ed\ d dub. B, EA y. 

24. f/jv^ om. V. BZ\ ZB BVat.v. s^&sta v. y.4%%- 
tog\ supra scr. m. 2 V. 



32 EUCLIDIS OPTICA. 

jclevQal aga avakoyov i6ovtai. l6tiv a^o, Ag ^ EZ 
XQbg riiv ZB^ ii AA JCQog xiiv AB. &XX* 6 r^g EZ 
stQog tijv ZB l6yog i6rl yvaQc^og' Ttal 6 rfjg AA 
aQa ^Qog rijv AB X6yog i6rl yv6Qv^og, Tcai i6ri xal 
h ro AB yviOQL^ov. xal rb AA a^a yvfOQiiidv i6riv. 

%a\ 

Tb Sod^hv ^fjxog imyv&vai^ TtrjlLxov i6rCv. 

l6rc} rb Sod^lv fiTixog rb AB^ i^^a S\ l6ra) rb T, 

xal Seov l6rc3 rb AB iifixog yv&vaij Jtrjkixov i6rCv. 
10 jCQo67a7Crir<o6av axrtvsg al FA^ 

FBj xal sllTlq^d^cs iyybg roi) 

S^^arog rov F iTcl rf^g axrl- 

vog rv%bv 6ri^Elov rb A^ xal 

iiX^(o Sca rov A ^rjfieCov rfj 
16 AB jtaQcckXrjXog sid^sta ij AE. 

iTCsl ovv rQLydivov rov ABF ^l 

scaQa ^Cav r&v tcIsvq&v ri^v BA 

^xrav ri AEj l6riv «pa, &ig fi FA TCQbg ri^v AE^ ovrwg 

il FA TCQbg ri^v AB. alk^ 6 rf^g FA TtQbg ri^v AE Myoq 
20 i6rl yvGiQLiiog' xal 6 rfjg AF aga TCQbg rijv AB kdyog 

yv(hQL^6g i6riv. xal yv(hQifi6g i6riv ij AF. yv(bQi[iog 

aQa xal ri AB. 

x^\ 

^Eav iv r& airp imTCsSo)^ iv cJ ro Sftfta, xvxkov 
26 TCSQLipsQSLa rsd^^^ ij rov x^ixXov JCSQLfpiQSia sifd^sta 
yQa^liil (paCvsrau 

i6r(o x^ixXov TCSQKpiQSia ij BF iv r^ airp iniTtiSGi 
xsLiisvrj r& o[i[iarL r& A^ atp^ oi 7CQ067Cinrir(o6av 

1. Ante ^mr del. comp. apa B. 4. xa^ (alt.)] om. 

BVat.v. 5. iaxi Vat. 6. xa'] xj3' codd. 9. %ai] om. v. 




EUCLIDIS OPTICA. 33 

ita da ad ah. sed e^ didi iz proportio est nota^ quia 
termini noti. quantitatis ergo da ad ah proportio 
nota. et est ab notum. ad ergo notum est. 

Datam longitudinem, quanta est, reperire. 

esto data longitudo ah^ oculus g^ et accidant radii 6 

ba et gh, et sumatur prope oculum g super radium 

forte punctus d, et trahatur per d punctum rectae ah 

parallela de recta. constituuntur tri- 
goni similes. uel sic. super ah magni- 
tudinem ab oculo ducatur cathetus dh, 10 
super dh autem adaptetur perpendi- 
cularis, donec per eius terminum e 
transiens uisus ueniat ad terminum h 
longitudinis cognoscendae. erunt igitur 

duo trigoni similes, et latera proportionalia, et pro- 15 

cedatur sicut prius. 

In eodem plano, in quo oculus, circuK periferia 
ponatur, ea circuli periferia recta linea apparet. 

esto periferia circuli hg, in eodem plano iacens 
oculus a, a quo accidant radii ah, ad, ae^ az^ ai, at, ag. 20 




9. uel sic] compendia duhia B. 10. ab] ad D. 12. 
per] pars.^ JD. Figtiram, quam dedi, praeter eam, quam codd. 
Graeci praebent, habet D. 



iatlv] iatl vm, et Vat., sed corr. 17. t&v TtXsvg&v] ti\v 
nXevQoiv m. 21. iattv (jpT.y] ieti YsLt.ym. 23. x|S'] xy' codd. 

24. xvxXov] comp. Vat.m, o B, o v. 26. xvxXov] comp. Bj 
ZXov Vat., corr. m. 2. 

Snclides, edd. Heiberg et Menge. YIL % 



34 



EUCLIDIS OPnCA. 




&xttv£g at AB^ AJ^ AE, AZ, AH^ AB^ AT. Uy&^ 
oxv ii BF neQiq>£QSia eid^sta q)aivstai. xs^tf^^m vilg 
nsQctpsQSLag tb xsvtQOv xal i6t(o to K^ xal isu^iBiix^^cih- 
6av svd^stav al 

6 KB, KA, KE, 
KZ, KH, KS, 
KF. ijtsl ovv 
^1 KB imo t7]g 
'bjtb KAB yo- 

viag ^XsTCstai^ 
il S^k KA 'bicb 
trig i)7cb KAA^ 
ILsClcDv aga q^av^^stai rj fisv KB tfjg KA^ ii 8s KA 
trig KE^ ij Ss KE tfig KZ^ xal ix tov stsQOv ^iQOvg 

5 ^ ^hv Kr trig K&, ^ Sh K0 tfig KH, ^ dh KH 
t7]g KZ ^si^fov q)avTfi6stai, Sia tovto Sii trig (isvoii^rig 
svd^siag tfjg KAf xdd^stog ij BF dsl i6tiv. td S* ait& 
6vfip7]6stai xal iTtl tfjg xoikrjg TtSQLfpsQsiag. 

"Akkog. 

Avvatbv Sh xal iit^ avtG)v t&v btl^scjv tavta Xiysvv^ 
oti ika%i6tri iihv i] [LStai^v tov A b^^atog xal tf^g 
Sia^stQOv^ dsl S^h ij iyyiov aitfjg iXdttaov tfjg djcA- 
tSQOv, taitd Sh 6v[L^aLVSi xal [idv^ xad^itov hC 
avtijv ov6rig tfjg AZ, Sid tovto (pavtaj6iav sifd^siag 

5 djco6tskXsL ij 7CSQLq)SQSLa^ xal fidXL^ta si dxb nksiovog 
q)aivoLto SLa6tiJiiatog &6ts fti^ ^vvaL^d^dvs^d^aL iiiiag 
tfjg xvQtdtrjtog. S^d tovto xal ol ^ij Tcdvv dTCotsta- 
fiivoL xdXoL ix Tckayiov fihv bQ&^svoL iy%dka6iLa 1%slv 



2. BT] FB Vat. 5. KB] BK m. 6. KZ] KF Bv. 

8. JS:B] BK y. 9. vTfd] supra scr. m. 2 V. KAB^ KB 



EUCLIDIS OFnCA. 35 

dico, qnoniam bg periferia recta apparet. iaceat peri- 
feriae centrum sitque Jc, et coniungantur Jcb, Tcd, ke^ 
TcZj Tciy Tcty Tcg. quoniam ergo Tcb sub angulo Tcab 
uidetur et Tcd sub angulo had, maior ergo apparebit 
Tcb quam Tcd et lcd quam Tce et Tce quam Tcz, et ez i 
altera parte Tcg quidem quam kt et kt quam ki et ki 
quam kz maior apparet. et propter hoc punctus 
plus uidetur appropinquare ad centrum quam e punctus j 
et e quam d ei d quam b. quare in apparentia uisus / 
aliquid tollitur de eius conuexitate. 1( 

Aliter. possibile est autem et in ipsis uisibus 
eadem dicere. quoniam enim minima quidem^ quae 
inter a oculum et diametrum, semper autem appro- 
pinquior ei minor ea, quae longius, ista uero con- 
tingant et catheto super eam existente a0j propter li 
hoc phantasiam rectae emittit periferia, maxime quae 
a plure apparet spatio, unde conuexitatem non per- 
cipimus. propter quod non multum extentae cordae 



3. Ante kab ras. 1 litt. D. 9. quare] in ras, D, in mg. qi. 
13. approquior D. 16. eroittet D, sed corr. 



Bv, et Vat., corr. m. 2. 12. v^6] om. Vat.^m. 13. fist- 

iov V. 16. (letSov V. 17. rijs] om. v. Tidd^etog] corr. ex 
%a&6tov V; lacuna est. i6Ti B vat.vm. 19. &XXoi)g] postea 
add. V, om. m. 20. x^' add. VVat. 21. Post Sti del. ^ V. 

A] te A BVat., ts v. 22. ^yiov] corr. ex ^yysiov V. 

iXdtt(ov] corr. ex fisi^oDv m. 2 V. icjtiiitsqov] icit6tsqov V. 

23. idv] om. BVat.v. 26. icjtocrsXsZ m. %aL] om. v, 

m. 2 Vat. 27. xv^r^TT^og] primum t in ras. V, xvQtdri\Toq 

Bv, et Vat., sed corr. ol] ij v. 



( 




36 EUCLIDIS OPTICA. 

Soxov6lv^ {moTcdtcad^ev d' eid^stg alvai^ ocal at 6xud S^ 
r&v xqCk(ov iv tp avt& imxiSfp xeviiivmv r^ (pcni- 
^ovrv Bi^slai yCvovrai. 

^'AXXcDg, 

6 ^Eav iv r« avrip ijtLTtiScj rip ^ftftart xiixkov mqi- 

(piQeuc rsd^^ ev^sta yQa^iiij fj rov xvxkov nsQKpiqsia 

q>aCvsrai, 

i6r(o xvxXov nsQi- B- 

(piqsia i\ BT^ &^iia Sh 
i6ra) rb A iv ra aiJrc5 

ixixiSfp ov rfiBr jcsqc- 

(psQsCcc^ acp^ oi 7Cqo6- 

ncjtrirca6av 'orl^scg al ^B^ -^Z, ^P. oixovv^ iTCScSij 

r&v 6Qa)fiivov oiShv oXov a^a bQarav^ sid-sta &Qa 
5 i6rlv fi BZ. 6[ioCG)g Sii xal ri ZF. Slrj aQa ^ BF 

jtSQcq>iQsca si^d^sta S6^sc. 

xy'. 

Z(paCQag dTto^SrjTtorovv dQco^ivr^g vjtb svbg biifiarog 
iXa66ov asl fjfic^^pacQCov (paCvsrac^ avrb S% rb SqA- 

50 fisvov rrjg 6(paCQag [liQog xiixkov nsQC(piQSca (paCvsrac. 

l6rc3 6(patQa^ f^g xivrQOv fihv rb A^ '6^(ia Sh i6ra) 

rb B. xal iyts^s^dxd^G) ii AB^ xal ix^s^kri6%cD rb Sca 

rfjg BA iTtCnsSov. Ttoc/i^sc oiv ro^ijv x^ixkov. TtocsCrcD 

rbv r^0H xvxkov^ xal xsqI ScccfisrQOV ri^v AB x^ixkog 

55 ysyQcc^pd^o 6 FBA^ xal iyts^s^dxd^co^av sid^stac a[ FB^ 
BA^ AA^ AF. iytsl ohv ii[icxvxlc6v i6rc rb AFB^ 
d^d^ii ycavCa i6rlv ij inb AFB* b^oCcag xal i^ i)Ttb BjdlA. 

1. 8o%ov6i V. BvQ^Bvg] -d'stg in ras. V, sited^stg v, sit^sa 
Vat.im. 4. mtag] %s' V Vat.v(B?), &XXG}g rb aM Vat.im. 



EUCLIDIS OPTICA. 37 

ex obliquo quidem xiisae dimissionem habere uidentur, 
inferius autem recti esse. et umbrae quoque. 

Esto circuli periferia bg, oculus uero d in eodem 
plano ig periferiae, a quo accidant uisus db et dg. 
igitur quoniam uisorum nihil totum simul uidetur, i 
recta ergo est b^. similiter autem et zg. tota ergo 
bg periferia recta est. 

Sperae qualitercunque uisae sub uno oculo minus 
hemisperio semper apparet, eaque uisa sperae pars 
sub circulo contenta apparet. n 

esto spera, cuius centrum a, oculus uero &, et 
coniungatur ab, et educatur per ab rectam epipedum. 
faciet ergo sectionem circuli, et sit gd, et circa dia- 
metrum ab circulus describatur gbd, et coniungantur 
gb et db, ag, ad. quoniam semicirculus est agb, li 
rectus ergo est agb angulus. similiter autem et bda, 
rectae ergo gb et bd contingentes sunt per tertium 

1. uidetur D. 2. Post quoque spat. uac. 4—5 lin. D. 

5. nichil D. 12. coniungantur D. 15. -circulus — 16. 
rectus] in ras. D, seq. tus. 

10. Ifftro)] om. Vat.*m. 14. r&v dgtoiisvoDV^ rov oQCDfisvov 

Wat.v. 15. 3Xri . . . rj BF 7teQL(psQ6uc] ZXriv . . . rijv BF iisqi- 
(piqsiav Vat.^m, m. 2 V. 16. si^-sta] ei^^stccv VVat.^m. 

Sd^ei] scripsi; i^ei VVat.^m, iartv Vat.v, non liquet B. 17, 
xy'] xs' VVat.v. 18. ivdgimpra, scr. V. rov §v6s Vat.v et 
postea ins. B. 19. hXarrov Vat. 20. iieqog] del. V, om. m. 

x^xXov 'TteQifpiQeiu] corr. ex iifiLnvTiXLov (i6vov m. 2 V. 23. 
BA] AB Vat.v? %v%Xov comp. m. 24. r6v] r^ B; r6 Vat., 
corr. m. 2. TtvxXov] corr. ex x^x^ov m, om. Bv, m. 2 Vat. 

xvxXog] iregos x^xXo? m. 25. FBd] FBdA Vat.^m, m. 2 
VVat. 26. Bd] d e corr. V. Ad, AT] AF, Ad BVat.v. 

iGTLv Bv. AFB] corr. ex ATd V, ABF m. 27. '^(avUt\ 
&QCC BVat.v. iGrlv] om. BVat.v. «fj (^a\\.^j\ om. ^ . 



38 



EUCUDIS OPTICA. 




aC FB^ Bjd &Qa i^ajcxovxat. inB^eix^iQ ovv ^ P^, 

Tcal ijx^^ ^*^ ''^^^ ^ 6ri^siov xy fk/ xaQdllrjXog ^ H0. 

dQ^al aQa ai 

jtQog x(p K, iav 
6 SiitbBrKxQL- 

yc3vov iievov6rig 

tfjg A B ne^X 

riji/ d^dijv y(o- 

vCav tijv K 
.0 steQUvex^^v eig 

xb a{>tb TtaXiv 

d^oxata6tad7i , 

od^ev i^Q^ato q>e- 

Qe^d^ai^ 71 fihv B F xad'^ ev ^rjiieLOv iq>dil;etaL xfjg 
.6 6(pavQag^ i^ Sh KF 7tOL7]6eL trjv tofiijv xvxkov, xvxlov 

ftii/ aQa neQiipeQeia dfpd^Tj^etaL iv t^ 6(paiQcc. key(o 

Se^ Zti xal ekattov iiiii6(paiQL0v. i^tel ydQ ri[iLxvKXi6v 

i6tL tb HG^ tb r^ iXattov fi^LXvxXCov i6tCv. xal 

* bQataL {)7tb t&v B f, B^ dxtCv(ov tb avtb tfjg 6(paCQag 

10 ^eQog. ekattov aqa iiiiL6(paLQCov tb T^' xal inb t&v 

dxtCvcjv t&v BF^ B/d pXeTtetaL. 

xS\ 

Tov a^fiatog 7tQo6L6vtog tfj 6(paCQcc iXattov e6taL 
tb 6Q(Ofievov^ S6^eL S^ fiet^ov bQcc^d^aL. 
15 e6tc3 6(paLQa^ ^g xevtQOv fihv tb A^ Sfiiia Sh tb B, 
d(p^ oi i^te^evx^cj eid^eta ii AB. xal iteQLyeyQa^p^^Gi 
neql tijv AB xvxkog 6 FBA^ xal ^x^^ ^^^ '^^'^ ^ 
6rjfieCov tfi AB ev^^eCcc itgbg dqd^dg i(p* exdtega eifd^eta 

1. BJ] corr. ex J B. ovv] om. BVat.v. 6. BFK] BKV 
BVat.v. 1, AB] KB m, m. 2 Vat.v. 13. (palQse^ai m. 



EUCLIDIS OPTICA. 39 

EuclidiS; scilicet quando a termino ducta existens 
linea facit angulum rectum, illa contingens erit. con- 
iungatur gd^ et trahatur per a punctum rectae gd 
parallela it recti ergo qui ad Tc auguli. si autem 
hgh trigonus manente ab circa rectum angulum h 5 
circumagatur, in idem rursum, unde incepit, feretur, 
et hg quidem unumquodque sperae punctum continget, 
hg uero faciet sectionem circuli. circuli igitur peri- 
feria uidebitur in spera. dico, quoniam et minus 
emisperio. quoniam enim semicirculus est ity gd minus 10 
semicirculo est. et uidetur sub hg radiis eiid eadem 
sperae pars. minus ergo hemisperio gd. et sub radiis 
bg et hd uidetur. 

Oculo accedente propius sperae minus erit, quod 
tddebitur, uidetur autem magis uideri. 15 

esto spera, cuius centrum a, oculus autem 6, a quo 
ducatur recta ab, et describatur circa ab circulus gbd, 
et trahatur ab a puncto rectae ab ad rectos punctos 
in utraque recta 60, et educatur quidem per e^ et ab 



1. Euclidis] -is in ras. D. 2. Ante erit del. enim est D. 
coniungantur D. 8. uero] mg. m. 1 D. 10. emisperia D. 
11. semicirculo] -o in ras. D. Post bg est — in ras. 2 

litt. D. 13. hd^ mut. in dh ml ah D. 



14. Br] BN V. 15. t^qv] om. BVat.mv. 16. tisv] om. 
BVat.v. rjf] om. codd. 17. $i] om. BVat.v. iTtsL — 
20. rj]'mg. m. 1 V; in textu est insl yccg i^iLtivxXLOv i6xi 
rb rjy postea expimctum. 18. H@] K0 m. iariv ijfirt- 

KvaXlov m. iariv] iaxL Vat.v. 19. BF] BN y. 20. 

rjiLLa(paiQiov v. t&v] om. BVat.v. 22. %d'] xj' VVat., 

eras. v. 23. t^] ^yiov rj Vat.v, postea ins. B. 25. i/biv] 
om. BVat.v. 27. AB] corr. ex AFY. 28. s^%d^^^, 

BVat.v. 



26 EUCLIDIS OPTICA. 

iX&66Qvi ^sttov rb i^sQfpaLvdfievov (pa(v€tai^ &jtL6vtog 
Se [iBL^ovt [jist^ov^l. 

"06a &XXTfik(av vnsQixBi^ i%^ sid^eiag ta iX&ttovi 
5 [isyid^si tov (i[L[Latog 7CQo6v6vtog ts %al &<pL6ta[iivov 
tp t6(p &sl S6^si th 'b7CSQ(paLv6iLsvov tov iX&ttovog 
'b%SQi%siv. 

i6t(D Svo avL6a [Lsyid^rj ra ^B^ P^, c)v [ist^ov tb 
AB^ HfHia Sh i6t(o tb Z ijc' sv^sCag xsl[isvov tp ni- 
10 QatL tov r^ [Lsyid^ovg tp F. ^ 
kiycDj otL tov Z oii[Latog 
3tQ06L6vtog xal &(pL6ta[iivov 
iit^ siy^siag Hvtog t& l6(p 
86i,SL 'b%sQ(paLvs6%'aL tb AB 

15 tOV T/d. %Q06%L7ttit(0 y&Q 

&xtlg SLa tov F 'fj ZE. tb 

AB ccQa rov JTk/ 'b7CSQ(paCvstai ta AE. [istaxsxLVil^d^cj 
Sii ro 6[iiia xal i6t(o &7tc3tiQ(o xal i6t(o iit^ sid^sCag 
ro H. 'fj ccQa &7tb tov H (iii[iatog &xtlg 7tQo67tC7ttov6a 
20 iXs^v^stac S^a rov T ^'riiisCov xal 7tQ06svsxd"^6staL [lixQc 
tov E 6i]iisCov^ xal rc3 a^bt^ 'b7tSQ(pav7]6staL tb AB 
tov TA. 

Lti . 

Tb Sod^sv vilfog yv&vaL^ ^friXCxov i6tCv^ 'fiXCov (paC- 
25 vovtog. 

l6t(o ro Sod^sv vil^og tb AB^ xal Siov a^btb yv&vaLj 
TfYiXCxov i6tCv. ^6t(o [ihv 6[nia ro -^, 'liXCov Sh &xtlg 



E F g ^ 



2. ftfrfov] om. Vat.^mv, m. 2 Vat. Dein add. '^il^? V, 
m. 2 Vat. 3. tj'] iri' codd. 6. tc5 hoi icBil in ras. m. 1 v. 



\ 



EUCLIDIS OPTICA. 27 

accedente ergo minori minus superapparens apparet, 
abscedente uero maius. 

Quaecunque altemorum se superant, in directo 
minori quantitati oculo accedente et abstante aequali 
semper uidebitur superapparens minorem excedere. 6 

sint duae inaequales magnitudines ah et gdy qua- 
rum ah maior, oculus uero sit z in directo iacens 
termino quantitatis gd ei, qui est g. dico, quod 
puncto g oculo accedente et abstante in directo exi- 
stente aequali uidebitur superapparens ah ei, quod 10 
est gd. accidat enim radius ie per g. itaque ah ei, 
quod est gd, superapparebit eo, quod est ae. trans- 
moueatur autem oculus et sit longius et sit in directo 
ei, quod est i. ab oculo ergo radius accidens ueniet 
per g punctum et adiungetur usque e punctum, et 15 
eodem superabit ah quidem gd, 

Datam altitudinem cognoscere, quanta sit, sole 
apparente. 

esto data altitudo ah, proponaturque eam cognoscere, 
quanta sit. sit oculus d, solis autem radius ga con- 20 



1. minori minus] in ras. D. 2. maius] in ras. D. 3. 
se] supra scr. D. 11. accidit D. ie\ scr. ze. 13. 

directe D. 



iXdeeovog Vat.v, comp. B. 9. B^bd-slagl -ccg in ras. v. 

TflS] tov m. 11. tov Z] tb Zm.l V, t^ r tov B Vat.Vat.^mv, 
m. 2 V. 15. ydg] om. m. 16. ZEl EZ m. 18. tb 6V^a] 
rov d^niatog v. &7totsQG} VBv, et Vat., corr. m. 2. ^ffra)] 
v,sic%'a} m. 19. t6\t& m. H (alt.)] om. Bv, m. 2 Vat. 

20. 8ia\ xal Sid B Vat.v. 23. trj'] i&' codd. 24. i(Stiv\ 
iatl tov V. 27. iatl v. 



28 



EUCLIDI8 OPTICA. 




I 



fj FA 6vfipdllov6a tp TciQaxi tov AB ^sye^ovg xal 

SvtIx^co ^i%Qi tov /1 ^[ifiatog. l6t(o Sh 6md ii AB 

tov AB. xal xe^^d^co 

€tSQ6v tL [leysd^og tb EZ 
5 6vfipdlkov tfi dxttvi ^ij 

TtdvtcDg Tcatavya^dnsvov 

vit* avtTJg xatd ro Z Tts- 

Qag. 7lQiio6taL ovv sig 

tb ABA tQiycDvov ets- 
10 q6v ti tQiycovov tb EZA. l6tLV aQa^ hg fi AE 

XQbg tYjv ZE^ ovtcDg fj AB ^Qbg f^v BA. dXV 
^- 6 tTjg AE Ttgbg tijv EZ k6yog s6tl yvAQLnog* xal 

6 tfig AB aQa TtQbg ti^v BA k6yog i6tl yvmQL^og. 

yvd}QLfiov Sh tb AB. yvd)QL^ov aQa xal tb AB. 

16 , Ld'\ 

Mii 'bTtdQjjovtog ijlLOv tb Sod^hv ii^og yvcbvaL^ nrj- 
kixov i6tiv. 

s6tG) tL Ijisyid^ovg^ vtf^og tb AB^ gftfio; Sh i6tco 

tb P, xal Siov s6ta) tb AB yv&vaL^ utrjlixov i6tiv^ 

20 6}g ^ij ifTtdQxovtog ijliov, xsi^d^CD xdtOTttQOV tb AZ^ 

xal jtQO^SK^s^XTf^^^-m tfj EA i7t* svd^siag ij AB^ ^XQi^g 

oi 6vfiPalsL t(p TtsQatL tov AB ^syid^ovg tp Bj xal 

nQ067tL7ttita) dxtlg aitb tov S^^atog tov F ij FH^ 

y xal &vtavaxsxXd6%'a)^ &X9^S ov 6viL^aksl t^ TtiQatL 

25 roi) AB ^syid^ovg tp A^ xal ^Qo^sx^s^kifi^^^G) tfi AE 

il ES^ xal ilx^^ ^^o tov F i^tl tiiv ES xdd^stog ij 

7. Ante Ticctd add. SclXd m, m. 2 VVat. 8. TjQ^ida&a} m. 

9. ABJ] corr. ex ABF Y. 10. z/E] z/Z Bv, et Vat., corr. 

m. 2. 12. JE] dZ Bv, et Vat., corr. m. 2. EZ] in ras. V, 

ZE BVat.v. 14. Post AB add. :<^ k^fjg V, m. 2 Vat. 15. 

t-O"'] %' codd. 17. iati v. Dein add. Igflff B, sed del. 



EUCLIDIS OFnCA. 29 

cidens termino a magnitndinis et protrahatur nsque 
ad oculom. sit autem umbra di altitudinis ab, iaceat- 
que altera quantitas e^ concidens radio non omnino 
illuminata ab eo secundum ^ terminum. aptatus est 
ergo ut abd trigono alter trigonus e^d. est ergo sicut 5 
de ad ^e, ita db ad ba. sed de ad eis proportio est 
nota. et db ergo ad 6a proportio est nota. notum 
autem db. ergo et ab, 

Non existente sole datam altitudinem, quanta sit, 
cognoscere. lO 

esto altitudo ab, oculus uero sit g, et sit proposi- 
tum ab cognoscere, quanta sit, sole non existente. 

iaceat speculum d^, et 
adiciatur rectae ed m d 
puncto db. terminus 15 
eius coniungatur ter- 
mino quantitatis ab, qui 
est 6, et accidat radius 
ab oculo g gi^ et re- 
fringatur terminus eius et coniungatur termino a ab ^O 
magnitudinis, et adiciatur rectae de recta et tra- 




5. ut] scr. in. 6. de {utrumque)] e in ras. D. 13. Supra 
speculum add. planum D. 19. oculio D, sed corr. 20. a] 
corr. ex ab D. 21. recta] corr. ex recte D. 



18. ftcy^^^og m. 19. iati v. 20. mg\ om. Bv, m. 2 Vat. 

tov riXiov m. 22. 6v^^ciXt[i V, sed corr.; avnpdXXij BVat.v. 

Post B del. lial TtQoasnPs^XT^ad^G} rfj JE i^ E0 B. 24. &va- 
TisyiXdad^a} B et Vat., sed corr. ; &voc7iS7iXrja&a} v. aviipaXy V, 
sed corr.; avyi^^dXXTi BVat.v. 26. AB'] corr. ex z/B V. 26. 
'h (P'^-)] supra scr. Y . 



{ 



30 EUCLIDIS OPTICA. 

r@. i^al oiv XQo67tB7tta)X£v ixtlg rj FH %al &vtava- 
xexla6tat 'fj HA^ TCQog l'6ag ycovCag avaxexka^iisvac 
sl6Cv^ hg iv totg KatOTCtQixolg kiyetai' l!6rj aQa yoDvCa 
fl {fTtb rne tfj ixo AHB. aXU xal fi ijtb ABH 
5 tfi inb rSH t^ri* xal XoLxii &^a fi inb HT® kovTCfi 
tfl ijtb HAB i6tiv l6rj. l6oy6viov a^a i6tl tb AHB 
tQCyovov tp FHG tQiy(bvG). t&v S\ C6oy(ovC(ov tQL" 
y(hv(DV avdloydv s16lv al utlsvQaC. S6tLv aQa^ iog ii 
rS JtQbg tiiv SH^ ovrcog fi AB ngbg tijv BH. aXV 
10 6 trig FG ^Qbg tijv 0H Idyog i6tl yv(DQL[iog' xal 6 
trig BA aQa ^jtQbg tijv BH l6yog i6tl yv^Qv^og. akX^ 
fl HB i6tL yvmQLiiog. xal fi AB aQa i6tl yvdiQLiiog. 



X , 



Tb Sod^sv pdd^og yv&vaL^ TtrjlCxov i6tCv. 
16 ^6tcj tb Sod^hv pdd^og tb AA^ Hiifia Sh l6t(o ro E^ 

xal Siov tb fidd^og yv&vaL^ TtrjkCxov i6tCv. 7tQ067tL7ttst(o 

yccQ tfj otl^SL fiXCov dxtlg rj 

EA 6viL^dXlov6a t^ im- 

TtiSip xarc!; tb B ^rjfistov 
20 xal t^ pdd^SL Tcatd tb A. 

xal TtQO^sx^s^kri^d^cD d^tb 

tov B ijt^ sid^sCag r] BZ^ 

xal iix^^ ^^^ ^oi) E iitl 

tiiv BZ sifd^stav xdd^stog 
26 fi EZ. i^tsl oiv t6rj ycjvCa fi ixb EZB t^ i^b BAAj 

&XXd xal i^ i)7tb ABA tfj ijtb EBZ^ xal fi tQCtrj aQa 

fj imb BEZ tfl i^tb AAB i6tLV t6r]. l6oy(ovLOv ccQa 

i6tl tb AAB tQCyovov t^ BEZ tQLy(ov(p. xal aC 

1. &va7iS'KXa6TaL Bv et Vat., sed corr. ; Scvrccvay.iTiXccTai m. 
4, rH(9] in ras. m. 6. XoLTtifj] XoLTtdvBy, Xom^] XoiTCoi t, 




EUCLIDIS OPTICA. 31 

hatur ab ocnlo g snper et cathetus gt quoniam 
ergo accidit radius gi et refringitur ia, ad aequales 
angulos repercussi erunt. aequalis igitur angulus t 
angulo iy et reliquus ergo reliquo. aequiangulus ergo 
tgi trigonus ail) trigono. est ergo sicut gt ad ti, 6 
ita et ah ad hL sed quantitatis g t a,d ti proportio 
est nota. ei ha ergo ad 6i proportio est nota. sed 
hi nota. ergo ei ha est nota. 

Datam profunditatem, quanta est, inuenire. 

esto data profunditas ad^ oculus autem sit e, sitque 10 
propositum cognoscere, quanta sit. accidat autem 
radius ed concidens plano ad punctum h et profundi- 
tati ad punctum d, et adiciatur a puncto h in directo 
h tjgr, et trahatur ab e super h0 cathetus e^. quoniam 
ergo et a anguli sunt aequales, et h contra se positi, 16 
erit et tertius tertio aequalis. quare trigoni similes. 
latera igitur proportionalia. est igitur sicut ez ad 0h, 



2. aequales] eaquales D. 4. i] eras. JD. 6. W] 

t B. 



n 

Xofc B. 6. HAB'\ ^ap m. td} tov v. 8. &Qa] supra scr. B. 

9. AB] AHBy et Vat., sed corr. 10. r&] TO Bv. yv&Qi-^ 
i^dg ieti BVat.v. 13. x'] %a codd. 14. ieti v. 16. 

th JE, xal 8bov] om. Vat., tb E ins. ante ^erro), xal 8iov post 
iata m. 2. 16. dsov htio Bv. i6ti Vat.vm. 17. rfl 

6>ft ijXiov] om. v, m. 2 Vat. 18. EJ] d dub. B, EA v. 

24. ti/jv] om. V. BZ] ZB BVat.v. sij&sta v. ^dd^e- 
tog] supra scr. m. 2 V. 



32 EUCLIDIS OPTICA. 

nksvQal &Qa avdloyov idovrai. l6ttv &Qa^ i^g ii EZ 
TtQbg xiiv ZB^ ii AA %qog tiiv AB. akk^ 6 r^g EZ 
TCQog riiv ZB kdyog i6tl yv^QLiiog' xal 6 tfig AA 
&Qa XQog tijv AB kdyog i6tl yvaQLiiog. xai i6tL xal 
h th AB yv^QLiiov. xal tb AA &Qa yvaQLiidv i6tiv. 

xa\ 

Tb Sod^lv iirjxog imyvcbvaL^ tctjIlxov ictCv. 

l6ta) tb dod^hv iifjKog tb AB^ Sft/xa 8^ i6t(o tb jT, 

xal Ssov £6ta) tb AB lifjxog yvcbvai^ jctjIlxov i6tlv. 
10 7CQ067CL7CtitG}6av dxttvsg a[ FA^ 

FB^ xal siki^q^d^cQ iyyi>g tov 

iililiatog tov F iicl tfig aml- 

vog tv^bv 6rj^£tov tb A^ xal 

i^Xd^co 8l& tov A 6rjii£L0v rij 
16 AB TcaQallrjlog ^vd^^ta ii AE. 

i7C£l oiv tQLyd)vov tov ABF ^i 

xaQd fiLav t&v 7cX£vqS)v tijv BA 

^xtaL ij AE^ i6tLV &Qa^ dyg ii FA TCQbg tijv AE^ ovrcog 

il FA TCQbg tijv AB. «AA' 6 tfjg FA TCQbg tijv AE l6yog 
20 i6tl yvd)QLfiog' xal 6 tf^g AF &Qa TCQbg tijv AB X6yog 

yvd)QLii6g i6tLV. xal yvd)QLii6g i6tLV ij AF. yvd)QL^og 

&Qa xal ij AB. 

'Edv iv t^ avt^ inLJciSG)^ iv o5 ro S/x/xa, xvxXov 
26 7C£QLq)iQ£La t^d^fj^ 'fj tov kvkIov 7C£QLq)£Q£La ^vd^^ta 
yQaiifiii (paLV£taL. 

l6tC3 XVXkoV 7C£QLtpiQ£La ij B F iv tp ait^ i7CL7Ci8G) 
X£L^ivrj tp HllliatL tW A^ d(p^ oi 7CQ067CL7Ct£t(x)6aV 

1. Ante ^ctiv del. comp. apa B. 4. xa/ (alt.)] om. 

BVat.v. 6. icti Vat. 6. xa'] xjS' codd. 9. xa^] om. v. 




EUCLIDIS OPTICA. 33 

ita da ad db. sed ez ad hz proportio est nota, quia 
termim noti. quantitatis ergo da ad ab proportio 
nota. et est db notum. ad ergo notum est. 

Datam longitudinem, quanta est, reperire. 

esto data longitudo a6, oculus g, et accidant radii 6 

ia ei gby et sumatur prope oculum g super radium 

forte punctus d, et trahatur per d punctum rectae ah 

parallela de recta. constituuntur tri- 
goni similes. uel sic. super ab magni- 
tudinem ab oculo ducatur cathetus db, 10 
super db autem adaptetur perpendi- 
cularis, donec per eius terminum e 
transiens uisus ueniat ad terminum b 
longitudinis cognoscendae. erunt igitur 

duo trigoni similes, et latera proportionalia, et pro- 16 

cedatur sicut prius. 

In eodem plano, in quo oculus, circuli periferia 
ponatur, ea circuli periferia recta linea apparet. 

esto periferia circuli bg, in eodem plano iacens 
oculus a, a quo accidant radii a6, ad, ae, az, ai, at, ag. 20 




9, uel sic] compendia dubia D. 10. ab] ad D. 12. 

per] pars.^ D. Figwram, qmm dedi, praeter eam, quam codd. 
Graeci praehent, habet D. 



i<stlv\ iatl vm, et Vat., sed corr. 17. t&v flrXfiv^coi/] x^v 
'jtXBvqdv m. 21. ^(yrM;(pr.)] ^ffrt Vat.vm. 23. xjS'] xy' codd. 

24. hvtlXov] comp. Vat.m, 6 B, S v. 26. xvx^ovj comp. Bj 
SXov Vat., corr. m. 2. 

SuolideB, edd. Heiberg et Menge. TLL ^ 



. ii 




34 EUCLIDIS OPTICA. 

dxttvsg at AB^ A^^ AE^ AZ^ AH^ A&^ AR kiytOj 
Sr^ il Br 3t£Qvq)6Q£vu sifd^sta ^aCvBxai. xev^d-c^ xf^g 
nBQi^BQBCag xh xivxQOv xal i6t(o tb K^ xal i7tB^Bvx^(o- 
6av Bvd^Btai at 
6 KB, KA, KE, 
KZ, KH, K@, 
KF. iicel ovv 
il KB inb rijg 
'bTtb KAB yco- 

10 vCag piixBtaL^ ^ 
rj d^ Kjd i^tb 
tf^g ijtb KAA^ 
fiBC^C3v &Qa q>av7]6Btac 'fj (ibv KB tfig KA^ ii 8b K^d 
tfig KE^ 'fj 8^ KE tfig KZ^ xal ix tov itigov ^iiQOvg 

16 fj filv KF tfig K@, fj 81 K@ tfjg KH, rj 8h KH 
tfig KZ iibC^(ov (pavfi^Btai, Slo: tovto 8^^ trjg ^BVOv^rjg 
Bifd-BCag trig KAf xdd^Btog 'fj BF aBC ictiv. ta 8' aitoi 
6viL^6Btai xal iitl trjg xoClrjg %BQi(pBQBCag. 

'AXX(og, 

20 Avvatbv 8b xal in' ait&v tmv Srl^BCjv tavta kiyBiv^ 
Ztv ila%C6tri ilbv fi ^Bta^v tov A 8ft/xaroff xal trjg 
8LafiitQ0Vj ocbI 8^ i^ iyyiov ainrjg ildttcjv trjg djt(o- 
tBQOV, tai)td 8\ ^v^^aCvBC xal [idv^ xad^itov i^C 
aiftijv ov6rjg tfjg AZ, Sid tovto (pavta6Cav B^d^BCag 

25 &no6tiXXBi 'fj nBQKpiQBLa^ xal fidh^ta bI aTtb nkBCovog 
(paCvoLto 8La6tfiiiatog &6tB fti^ ^vvaL^d^dvB^d^av fjiiccg 
trjg xvQtdtrjtog, 8Ld tovto xal ot ^ij Ttdvv dnotBta- 
(livoi TcdloL ix nlayCov [ihv 6q6iibvol iyxdkae^a ix^LV 

2. BT] TB Vat. 6. KB] BK m. 6. KZ] KF Bv. 

8. KS] BK Y. 9. 'b7t6] supra scr. m. 2 V. KAB] KB 



EUCLIDIS OFnCA. 35 

dico, qxLOniani hg periferia recta apparet. iaceat peri- 
feriae centrum sitque ife, et coniungantur Jcby hd^ Tce, 
kz, hi, Tct, hg, quoniam ergo lch sub angulo Jcah 
uidetur et M sub angulo Jcad, maior ergo apparebit 
lch quam Jcd et hd quam lce et lce quam 1c0y et ex 6 
altera parte lcg quidem quam Jct et lct quam Jci et lci 
quam lcz maior apparet. et propter hoc punctus 
plus uidetur appropinquare ad centrum quam e punctus j 
et e quam d et d quam 6. quare in apparentia uisus / 
aliquid toUitur de eius conuexitate. 10 

Aliter. possibile est autem et in ipsis uisibus 
eadem dicere. quoniam enim minima quidem^ quae 
inter a oculum et diametrum, semper autem appro- 
pinquior ei miaor ea, quae longius, ista uero con- 
tingant et catbeto super eam existente a^, propter 15 
hoc phantasiam rectae emittit periferia^ maxime quae 
a plure apparet spatio, unde conuexitatem non per- 
cipimus. propter quod non multum extentae cordae 



3. Ante kah ras. 1 Utt. D. 9. quare] in ra>s. D, in mg. q2. 
13. approquior D. 16. emittet D, sed corr. 



Bv, et Vat., corr. m. 2. 12. V7c6] om. Vat.*m. 13. fist- 

fov V. 16. (isiiov V. 17. Tfjg] om. v. ndd-STog] corr. ex 
wx&hov V; lacuna est. ioTi B vat.vm. 19. &XIods] postea 
add. V, om. m. 20. x^' add. VVat. 21. Post 8ti del. ij V. 

A] TS A BVat., Ts v. 22. lyyiov] corr. ex lyysiov V. 

iXaTToav] corr. ex (isLtoav m. 2 V. &7CtSoTSQOv] &yt6TSQov V. 

23. idv] om. BVat.v. 26. &7to6TsXst m. ticcL] om. v, 

m. 2 Vat. 27. Hv^rdrijrog] primum t in ras. V, nvQt^rrj^og 

Bv, et Vat., sed corr. oi] ii v. 




36 EUCLIDIS OPTICA. 

Soxov6lv^ {>noxdt(od'6v d' sid^stg bIvccl^ ocul al 6xLal 8h 
t&v XQLXODv iv tp uvt^ iTcmiSip ocsLiiivcov t^ fpfOtC- 
^ovtL sifd^staL yCvovtaL. 

"AXX(og. 

6 ^E&v iv tp avt(p inL7ciS(p t^ 'd^ifiatL xvxkov nsQL- 
(piQBLa t£%^^ ev^^eta yQafiiiri 'fj tov xvkXov nsQi^iQSLa 
(paCvBtaL, 

l6t(0 OC^xkoV %£QL- B- 

q>iQ£La 'fl BF^ b^iiia S^ 
10 icft(o tb A iv t& ait^ 

iTCLTtiSp hv t^BF 7t£QL- 
(p£Q£CcC^ atp^ oi 3tQ06- 

7tL7Ctit(X)6av '6ilj£Lg al jdB^ jdZ^ jdF, oixovv^ i%£LSii 
t&v bQ(D^iv(ov oi)S\v olov afia bQ&taL^ ^ifd^^ta HcQa 
15 i6tlv ii BZ. b^oCcDs Sij xal fj ZF, olrj ccQa fi BF 
7t£QL(piQ£La edd^^ta S6^£l. 

xy\ 

2Jq)aCQag bTtcjeSrjTtotovv 6Q(oiiivrjg iito ivbg Sft/tarog 
ika66ov a£l fiiLL6(paLQCov (paCv£taL^ aitb S^h tb 6q(o- 

20 fjLBvov trjg 6(paCQag fiiQog xiixlov n£QL(piQ£La q)aCv£taL. 

i6t(o 6(patQa^ fjg xivtQOV ^hv tb A^ S/tfto; S\ §6tc3 

tb B. xal i7t£^£vx^f^ ^ -^-S? ^l ixp^pXi^ed^a) tb Slu 

tfjg BA i%C%£S6v. 7tOLij6£L o{>v ro/LMji/ xvxXov. TtOL^Ctca 

tbv rA@H xvxlov^ xal n£Ql SLdii£tQOv tijv AB xihckog 

26 y^yQd^d^o} 6 FBA^ Tcal iTt^^T^x^CD^av ^xfd^^taL aC FB^ 
Bjd^ A^d^ AT. i%£\ ohv ijfiLXvxkL^v i6tL tb AFB^ 
dQdij ycDvCa i6tlv i^ i)7tb AFB' b^oCcjg xal ii {)jtb BAA. 

1. 8ovLo^<SL V. Biy^Blg] -d-sZg in ras. V, s^bad-stg v, si&sa 
Vat.»m. 4. &XXaig] xe' VVat.v(B?), &XXaig rb aM Vat.^in. 



EUCLIDIS OPTICA. 37 

ex obliquo quidem uisae dimissionem habere uidentur, 
inferius autem recti esse. et umbrae quoque. 

Esto circuK periferia hg, oculus uero d in eodem 
plano ig periferiae, a quo accidant uisus db et dg. 
igitur quoniam uisorum nihil totum simul uidetur, 5 
recta ergo est iz. similiter autem et 0g. tota ergo 
hg periferia recta est. 

Sperae qualitercunque uisae sub uno oculo minus 
hemisperio semper apparet, eaque uisa sperae pars 
sub circulo contenta apparet. 10 

esto spera, cuius centrum a, oculus uero 6, et 
coniungatur ah, et educatur per ah rectam epipedum. 
faciet ergo sectionem circuli, et sit gdy et circa dia- 
metrum ah circulus describatur ghd^ et coniungantur 
gh et dh, ag^ ad, quoniam semicirculus est agh, 16 
rectus ergo est agh angulus. similiter autem et hda. 
rectae ergo gh et hd contingentes sunt per tertium 

1. uidetur D. 2. Post quoque spat. uac. 4—5 Un. D. 

5. nichil D. 12. coniungantur D. 15. -circulus — 16. 
rectus] in ras, D, seq. tus. 

10. IffTO)] om. Vat.^m. 14, t&v dQatiivGov^ tov 6q(0(isvov 

Wat.v. 15. 8X73 . . . 7j Br jcsQitpsQSia] SXriv . . . tijv BF jcsqI" 
(pSQSiav Vat.^m, m. 2 V. 16. si&sZa] siO^stav VVat.^m. 

86^sC\ scripsi; IJtt VVat.^m, iativ Vat.v, non liquet B, 17. 
xy'] xs' VVat.v. 18. Irdfft supra scr. V. tov kv6g Vat.v et 
postea ins. B. 19. bXattov vat. 20. fi^f^og] del. V, om. m. 

x^xXou ^SQKpiQSia^ corr. ex ijfiiTivTili^ov \i,6vov m. 2 V. 23. 
BA] AB Vat.v? tivtlXov comp. m. 24. t6v] tb' B; t6 Vat., 
corr. m. 2. %v%kov'\ corr. ex yL'6ytXov m, om. Bv, m. 2 Vat. 

HvxXoff] ^tSQog nMog m. 25. TBA] FBJA Vat.*m, m. 2 
VVat. 26. BJ] J e corr. V. AJ, AF] AT, Ad BVat.v. 

iativ Bv. ATB] corr. ex ATJ Y, ABT m. 27. yavla] 
aQa BVatv. iatlv] om. BVat.v. «f^ (^\t.^\ qtesi, ^ , 



38 



EUCLIDIS OPTICA. 



at FB^ B^ &Qa iq>dntovtav, ins^siix^f*^ ovv ij J!^, 

xal iixd^a} Slo: tov A 6rj(i£vov tff F^ ^aQdklriXog fi H0. 

dQd^al &Qa ai 

TtQog t(p K. iav 
5 SiitbBrKtQL- 

y^ovov ii€vov6rjg 

tfig A B nBQl 

trjv dQd^v y(o- 

viav f^v K 
10 nsQLSvsx^^v sig 

tb aitb Ttdhv 

a7Coxata6tadij^ 

Sd^sv i^Q^ato (ps- 

QS^d^aL^ fj ft^T/ BF xad"^ sv erjiistov i(pdtl;staL tfig 
16 6(paiQag^ 'fj 8h KF TtoLr^^sL tijv to/lmJi; ocvxlov. xvxlov 

[ihv aQa nsQL(piQSLa o^pd^i^estaL iv t^ 6q)aiQa. ksycj 

Si^ StL xal sXattov 'fi^L6q)aLQiov. i^tsl yaQ ii^LXvxXLdv 

i6ti ro Jff@, ro FA ilattov i^iLXvxliov i6tiv. xal 

* bQ&taL vnb t&v BF^ Bjd axtivcav ro a-^ro tfig 6(paiQag 

20 fiSQog. slattov aQa iiiiL6(paLQiov tb F^' xal v%b t&v 

dxtivcov t&v Br^ Bjd ^ksnstaL, 




25 



x8\ 

Tov biiiiatog %Q06L6vtog tfj 6(paiQcc skattov s6taL 
tb bQo^iisvov^ S6^SL 8h iist^ov bQcc^d-aL. 

s6tca 6(patQa^ ^g xivtQov ft^i/ tb A^ ^iL^a Sh ro 5, 
d(p* oi iTts^svx^ca sifd^sta ii AB, xal TtSQLysyQd^pd^a) 
nsQl tfiv AB xiixlog 6 FBjd^ xal iix^^ ^^^ ''^ov A 
6ri^siov tfj AB sid^sitx ^Qbg dQd^dg i(p* sxdtSQa svd^sta 

1. BJ] corr. ex J B. ovv] om. BVat.v. 6. BTK] BKT 
BVat.v. 7. AB] KB m, m. 2 Vat.v. 13. (palqBG^oci m. 



EUCLIDIS OPTICA. 39 

^EuclidiS; scilicet quando a termino ducta existens 
liiiea facit angulum rectum^ illa contiiigens erit. con- 
iungatur gd, et trahatur per a punctum rectae gd 
parallela it recti ergo qui ad lc auguli. si autem 
hgh trigonus manente ah circa rectum angulum h fi 
circumagatur, in idem rursum, unde incepit, feretur, 
et hg quidem unumquodque sperae punctum continget, 
hg uero faciet sectionem circuli. circuli igitur peri- 
feria uidebitur in spera. dico, quoniam et minus 
emisperio. quoniam enim semicirculus est it^ gd minus 10 
semicirculo est. et uidetur sub hg radiis eihd eadem 
sperae pars. minus ergo hemisperio gd, et sub radiis 
bg ei hd uidetur. 

Oculo accedente propius sperae minus erit, quod 
uidebitur, uidetur autem magis uideri. 15 

esto spera, cuius centrum a, oculus autem 6, a quo 
ducatur recta ah, et describatur circa ah circulus ghdy 
et trahatur ab a puncto rectae ah ad rectos punctos 
in utraque recta e0, et educatur quidem per ez et ah 



1. Euclidis] -is in ras. D. 2. Ante erit del. enim est D. 
coniungantur D. 8. uero] mg. m. 1 D, 10. emisperia D. 
11. semicirculo] -o in ras. D. Post hg est — tn ras. 2 

litt. D. 13. hd} mut. in dh uel ah D. 



14. BT] BiNT V. 15. fqv] om. BVat.mv. 16. ftev] om. 
BVat.v. rj] om. codd. 17. 8b] om. BVat.v. iTtal — 

20. Pi^]'mg. m. 1 V; in textu est insl yaq inLiY,vyLli6v iezi 
tb rj, postea expunctum. 18. H&] K@ m. iexiv rjiH" 

TiVTLXiov m. iatlv] iati Vat.v. 19. BT] BN y. 20. 

iHLLGtpaiqiov V. t&v] om. BVat.v. 22. %8'] xf VVat., 

eras. v. 23. rg] ^yyiov rj Vat.v, postea ins. B. 25. ftfiV] 
om. BVat.v. 27. AB] corr. ex AF V. 28. siy^Bia] om. 

BVatv. 



40 EUCLIDIS OPTICA. 

^ jBZ, xal ixps^kTi^d^cQ tb Slo: t&v EZ^ AB ixCnsdov. 

3tOL7l6€L OVV tO^ijV TcdxloV. l6t(0 6 FEZjd^ Ttal iTtS' 

i€vx^(o6av aC FA^ A^^ AB^ BF^ T^. Slcc di) ro tcqo 

aitov iQ^^al [fi^v] at jcqos totg F^ ^d 6riiieL0ig. itp- 

5 dmovtav aQa aC BF^ B^^ aitivig s16lv axtlvsg^ Tcal 

pieTtetac ino tov B 'd^^atog tb Fjd [liQog trjg 6q)aLQag. 

lietaxexiviled^a} Sij tb S/tfta lyyiov tfig 6(paLQag xal 

l6tG) tb ®5 ag>' oi ijte^e^^x^cj eifd^eta ij 0A^ xal [^f^O 

yeyQd^pd^cj xiixkog 6 AAK^ xal iTte^evx^cn^av al SK^ 

10 KA^ AA^ A® eid^etaL. biiOLcog Sij i)%b tov ® <i[i^atog 

♦ pkeTtetaL fihv tb KA ^eQog tijg 0(paLQag^ 'bnb S% tov B 

. i^keneto ro FA. ekattov S\ ro KA rot) FA. jtQ06- 

Ldvtog ccQa tov ^fi^atog iXatt6v i6tL tb bQ(b^evov. 

Soxet Sl [let^ov (paCveCd^aL' ^eC^cov y&Q ij ijtb K0A 

16 ycovCa trjg ijtb FBA ycnvCag. 



xe . 



HtpaCQag SlA Svo d^^dtcjv bQc^iievrjg i&v ii Svd- 
fietQog tfjg 6^aCQag t6ri ^ r^ evd-eCa^ i(p' r^v SLeetrj- 
xa6L toi a^^ata dit^ akkijkmv^ tb riiiL6^aCQL0v avtfjg 
20 6(p%ifi6etaL Zkov. 



2. ovv\ 8ri BVat.v. iitBtBvx&fo Bv. ^. dS] AB m. 

rj] om. Bv, m. 2 Vat. 5. agoc] in ras. V, Si BVat.v. 

slai V. 7. ^yyiov] corr. ex ^yysiov V. 8. iTtB^svx^oi 

sif&sta i\ 0A %ai] supra scr. m. 2 V. TCSQiysyqd^p^fo] TtSQi- 
supra scr. m. 2, supposita lineola, V. 9. Ante %vyiXog add. 

TtSQl ti\v ®A BVat.v. xvxXog] -avvXov v, O" B. AAK] 

AAGK m, m. 2 VVat. iTtstsvx^oi Bv. 10. sifd-slcc v, 

comp. B. &] supra scr. m. 1 v. 11. tov] tfjs BVat.v. 12. 
ipXsTts Vm. 15. FBJ] KBJ m. 16. tls'] %i\ V, xf Vat.v. 

18. ^J supra scr. m. 1 B. ^v] ^s BVat.v. 



EUCLIDIS OPTICA. 



41 



enipipedum. faciet autem sectionem circuli. esto gezdj 
et coninngantur ga, ad, dh, hg. per praemissum uero 
theorema rectae, quae ad g, d puncta. contingunt 
uero hg et hd, quae sunt radii, et uidetur sub 6 
oculo gd pars sperae. transmoueatur autem oculus 




5 



propius sperae et sit iy a quo ducatur recta tay et 
describatur circa ta circulus alJc. coniungantur tlc, lca, 
aly It similiter autem sub t oculo uidetur Tcl pars 
sperae. sub h uero uidetur gd. minor autem hl 
quam gd, accedente ergo oculo minus est, quod uide- 10 
tur, uidetur autem maius apparere. maior enim qui 
sub ktl angulus eo qui est sub ghd angulo. 

Spera a duobus oculis uisa^ si diametro sperae 
aequalis fuerit recta, in qua a se inuicem oculi distant^ 
emisperium eius uidebitur totum. 16 



3. rectae] mg. m. ID. 7.t a 
8. hV] corr. ex ki JD. 9. kt 



mg. m.lD. alk] corr. exaik D. 
kiD. l^Si. «bTi^QX?{ii\wga!!L^TIi< 




42 EUCLIDIS OPTICA. 

l6t(o 6^atQa^ ^g xivxQOv tb A^ xal yByQcc^d^Gi iv 
tfj 6tpaCQa scsqI xivtQOv tb A ocixkog 6 BF^ xal i^xd^Gj 
SiaiLStQog aitov ii BF^ xal iJx^(o<fav anb t&v 5, F 
XQbs dQd^d^s ccC 

6 B^, TE, tfi ds ^fl^ ,JF 

B r TtaQalXri- 
kog £6ta) ii jdE^ 
i(p^ '^g ocsi^d^ca 
tct Siiiiata rA 

10 A^ E. Xiyto^ 

Zti tb iiiLLetpaCQLOv oXov 6q)di]6£taL. i^x^fo SlA tov A 
STcatiQa t&v BA^ FE jtaQdlkrjlog ii AZ' tb ABAZ 
&Qa jtaQalkrjkdyQafifidv i6tLV, iav Sij iievov^rjg tfig 
A Z 7t£QL€V€x^^v £lg TO avtb Ttdhv &7toxata6tad^^ o^£v 

16 fiQi,ato q^iQ^^d^aL tb 7t£QL£V£X^hf 6xW^^ aQ^£taL ^lv 
aitb tov 5, ik^^de^taL Sh xal iTtl ro F xal tb 5, xal 
tb 7t£QLyQaq)hv i)7tb tfjg AB cf^^/xa xvxlog ietaL^ og y£ 
SlA tov xivtQOv tfig 6tpaCQag i6tCv. fj^L^^aCQLOv aQa 
6(p%"ifl6£taL i)7tb tS)V A^ E dfi^dtov. 

20 xs:'. 

'Eav ro td)V dfiiidtmv SLd6trj^a ii£t^ov ^ tfig iv tfj 
6(paCQa SLafiitQOv^ (1£l^ov rov r]iiL6(paLQCov 6(pd^6£taL 
tfjg 6q)aCQag. 

£6t(o 6q)aLQa^ '^g xivtQOv ro A^ xal Tt^QLy^yQd^pd^co 

26 7t£Ql xivtQov ro A xvxXog 6 E0AH^ Hfifiata Sh t& 

5, F, xal l6t(o tb SLd6tri^a ro fi£ta^v tmv 5, F 

brl)£C3V ^£L^ov tfjg iv tfj 6(paCQcc SLa^itQov^ xal i^t^- 



1. aqxxiQa] C^ m, ut alibi, 3. BT] BiNT v. T] 

N T. 6. bJ] /^ in ras. V. 9. ^fi-ora y. 13. Tta^QLllTilo- 



EUCLIDIS OPnCA. 43 

esto spera^ cuius centrum a, et describatur in spera 
circa centrum a circulus hg, et trahatur diametros 
eius hgj et trahantur a punctis h, g perpendiculares 
hd et gCy et rectae hg parallela esto de (et ita par- 
aJlelay quod perpendicularis ducta ab a puncto cadat 5 
super medium punctum de*^ aliter enim non esset 
uerum), in qua iacent oculi d et e. dico, quod totum 
hemisperium uidetur. trahatur per a utriusque hd 
et ge parallela az, itaqae dbdz est parallelogrammum. 
si autem manente az circumducatur in idem rursum, 10 
unde incepit, restituetur descripta figura, incipiet qui- 
dem a 6, ducetur uero et super g, et descripta quidem 
sub ah figura erit circulus, qui utique per centrum 
sperae est. hemisperium ergo uidebitur sub d, e 
ocuKs. 16 

Si oculorum distantia sperae diametro maior fuerit, 
plus hemisperio uidebitur. 

esto spera, cuius centrum a, et describatur circa 
centrum a circulus etdi, oculi uero 6, g, et sit spatium 
uisuum hy g intermedium maius ea quae in spera 20 



4. paralella jD, ut saepim, 6. esset] mg. D, in textu er 
€ corr. 9. paralellogramum D. 12. ?. 6] c corr. D. 18. 
circa] contra D. 20. spera] om. D. 



ygaiiiMv] TcaQaXXriXo* B, naQdXXrilog v, et Vat., corr. m. 2. ^i}] 
om. V. 14. AZ] corr. ex AH B. o&sv] %viiXog ^f v. 16. 
wxl tb B] om. Bv, m. 2 Vat. 17. iivyiXog] comp. BVat., 

o-^o V. og] mg v. 18. iati v, comp. B. 20. xs'] x^' V, 
XT]' Vat.v. 21. t6] supra scr. V. 22. tov] bis Vat., sed corr. 

tfjg OfpalQag dtp^rjastai, m. 25. TtSQi] ta Vat.v. 26. 

t6 (alt.)] tmv BVat. 



44 



EUCLIDIS OPTICA. 



6q)d"il6etav, nQo6nLntit(o6av ixttves al BE^ F^ xal 
TtQoeexPepXilad-cj^av inl ti^ E^ A ^iQrj' 6v^pdkkov6ir 
Sii &kkifilaig Slo: tb iXd66ova elvai ti^v SidiietQOv t^g 

6 BF. 6viiPakkit<o6av Sij xat&, tb Z oriiietov, ixel ohv 
an6 tLvog ^riybeCov t&v ixtbg tov xvxXov ^qos tiiv 
neQL^piQevav 7CQ06nent(hxa6LV evd^etaL al ZE^ Z^, tb 
jd®E a^a ilatt6v i6tLV inLLXvxkCov, tb EHjd aQcc 
liet^6v i6tLV ii^LxvxXCov, «AA' i)7cb t&v B^ F tb EH^ 

10 pkijtetaL. [let^ov a^a ^ tb il^L6v 6(pd"i^6etaL tov xi5xAov 
i)nb t&v By r. tb aiftb &Qa xal trjs 0(paCQas 6^)%"^- 
6etaL. 

x^\ 

^E&v tb t&v 6iiiidt(ov SLd6triiia iXattov fi trjg iv 
15 r^ 6(paCQ(f SLafiitQov^ iXattov fiiiLL6(paLQCov ^^pd^ij^etaL. 




i6t(D 6(patQa^ 'fjg xivtQov tb A ^rjiietov^ xal neQL- 
yeYQd(p^G} %eQl ro A ^rnnetov xiixlog b BF^ xal xeC^d^c} 
ro Svdetrjiia t&v 6iiiidt(ov tb AE ika66ov 8i/ trjg iv 



1. Bri BN V. 3. TtQOGSTipspXi/ia&a} VBVat.vm. crv^- 

paXovai BVat.v. 5. <!vy,§ccXXhai Bv et Vat., sed corr. 6, 

t&v] xov Vat. 8. J&E] e corr. V, :>' 0E Bv, d^ 0E Vat., 

sed corr.; /i@ Vat.^m. %XaG6ov BVat.v. t6\ r6 8i Vat.v. 

10. fisl^ov] om. V, m. 2 Vat. i]] om. v, m. 2 Vat. ^(iiav 



EUCLIDIS OFnCA. 45 

diametrO; et coniungatur hg, dico^ quoniam maius 
hemisperio apparet. accidant enim radii he et gd et 
educantur super e, d partes. concurrent uero adin- 
uicem propter minorem esse diametrum recta hg. con- 




cidant autem ad punctum z. quoniam igitur ab aliquo 
puncto extra circulum dato uidelicet z uidetur ad 
periferiam accedere ze et zd, semicirculo ergo minus 
est det maius ergo semicirculo eid. sed sub h, g 
oculis idem uidetur. itaque maius dimidio circuli uide- 
bitur sub h, g. idem ergo et sperae uidebitur. 

Si OGulorum distantia ea quae in spera diametro 
minor, minus hemisperio uidebitur. 

esto spera, cuius centrum a, et describatur circa a 
circulus hg, iaceatque spatium oculorum de minus 



1. coniungantur D. 5. igitur] gi (igitur) vsl go (ergo) D, 
qime compendm omnino difficulter dignosctmtur, uelut p. 47, 7, 
8. det'] corr. ex dez D. 9. idem] post ras. 1 litt. D; scr. 
eid. 13. circa] vn ras. D. 



x^xXoff V. 11. &qa aM v. 13. xj'] >t' V, >t^' Vat.v. 14. 
Tijg] e corr. Vat. 16. iXccaaov v, comp. BY^t» 



46 EUCLIDIS OPTICA. 

t^ 6^aCQa SLUfihQOv^ &q>^ cS i^x^fodav i^a%r6iuvav 
at jdB^ EF at ai^xal ocal dxttvBg. kiyto^ Zti ika66ov 
illit^q^aLQvov 6(pd^6£tai. i%§s^kTfi6%'(o6av y&Q aC Bjd^ 
FE' 6viixs6ovvtaL di^ ijtl rcfc F^ H^ B [liQfi, ^sLSi/pceQ \ 
5 fi jdE ikci66c3V i6tl tijg iv tfj 6q)aLQa SiafiitQOV. 6v(JL- 
7tL7Ctita)6av xat& tb Z ^rnistov. ijcsl oiv &7t6 tvvog 
6riiiSL0v tov Z nQ067Csm(hxa6LV sid^staL aC ZFj ZB^ 
tb BHr aQa iXatt6v i6tLV fniLXvxkLOv. c^AA' iv c5 
i6tL tb BHF t(i7j(ia^ iv tovtp xal tb trjg 6(paCQag. 
10 &nokaiipdvov6LV &Qa ikattov fifiL^^aLQCov, 

xrj\ 

KvUvSqov bxcD^Srpcotovv vnb svbg b^iiatog 6q(o- 
liivov ikattov iifiLXvkLvSQCov 6(p%^6staL. 

i6t(o xvkLvSQog^ oh i6t(o xivtQOV tfjg pd6s(og tb A 
16 6rjfistov^ xal TCSQLysyQciq^d^ca tcsqI tb A xvxkog 6 BFy 
xal xsC^d^co 8fi- 
fia tb A iv tp 
aiftca ijCLTciScQ 
xsCfisvov tri pd- 
20 6SL tov xvXCv- jj 

Sqov tfj Br^ 

xal iTCs^svx^G} 
aTcb roi) A inl 
tb A 'fi AA^ 
26 xal ¥jx%c:i6av aTcb rov A axttvsg aC AB^ AF^ xal 
i^aTCti^d^co^av rov xvxkov^ xal dvfix^^^av d%b t&v 
B^ r 6rifisCc3v TCQbg ^Qd^dg TcksvQal tov xvXCvSQ(yv aC 
BE^ rZ^ xal ix^s^kifi6%^(o rd ts Sl^ t&v AB^ BE 

2. %Xatxov Vat., comp. B. 3. iyipspXrja&a) Vat. v, comp. B. 
4. r, H, S] FBH m. iiitQri Vat., corr. m. 2. 5. ^Xaccov v, 




EUCLIDIS OPTICA. 47 

existens ea quae in spera diametro^ a quo trahantur 
contingentes db et eg et cedere et radii. dicam^ quo- 
niam minus hemisperio uidetur. educantur enim hd 
et ge. concidant autem in gih partes, quoniam qui- 
dem de minor est ea quae in spera diametro. con- 6 
cidant ad punctum z. quoniam ergo ab aliquo puncto 
uidelicet accidunt ^g et zb, igitur big minor est semi- 
circulo. sed in quo big sectio, in hoc et sperae. con- 
tinent ergo minus hemisperio. 

Cliilindro quaKtercunque sub uno oculo uiso minus 10 
hemicliilindro uidebitur. 

esto chilindrus, cuius sit centrum basis punctus a, 
et describatur circa a circulus bg, iaceatque oculus d 
in eodem iacens plano basi chilindri bg^ et coniungatur 
ab d super a recta da, et trahantur ab d radii db, dg 16 
et contingant circulum, et trahantur a punctis b, g 
ad rectos angulos latera chilindri be et gz, et educa- 
tur quidem per db et be ebipedum et quidem per dg 



2. cedere] ced'e D; scr, eidem. 5. eal e corr. D. 6. 
aliquo] e corr. D. 8. continet D. 14. chillindri D, ut alibi. 
15. et — 16. circulum] mg. m. 1 D. 18. &e] c corr. D. 



iXdTtav Vat., comp. B. 7. f^-O-ftat] om. BVat.v. ZB] 

rZB Y. 8. iXocaaov v, comp. B. 9. iari] om. BVat.v. 

BHT] BTH m. 10. iXaccov v, comp. B. 11. x?]'] 

Xcc' Y, X BVat.v. 13. ^Xocttov] comp. B, iXdaaova v. ijiu- 
xvXivdQiov] i}y,iiivXlv$Qov Vm, rjiiitivXivdQiov BVat.v. 14. 

TivXiv^Qogl Vat.^m, corr. ex li&voQ m. 2 VVat., n&vog Bv. 

foroi] om. m. §dcas(og'] comp. BVat., §ccarig v. arni,sZov 
tb A BVat.v. 25. rix^(o Vat. (corr. m. 2) v, comp. B. 

z/r] JN Y. 26. icpocTtTiad^o) Vat.v, comp. B. &vrix^a}' 
aav] icvrix^o) Vat.v, comp. B. 27. <rr\(i8i:a Y^A,..^ ^ort. tssl. 'L, 



48 EUCLIDIS OPTICA. 

iTCiTCBSov xccl rb Si,& r&v jdF^ FZ. oifSirsQOv &Qa 

air&v riiivBv tov xiihvSQOV iq)&Tttovtai yocQ xal ccC 

\ ^B^ ^r xal ocC BEj FZ. ^XijtBtai ohv ijtb r&v 

B^j ^r &xtCv(ov tb BF^ oJtsQ i6tlv ikatxov i^ftt- 

5 xvxXlov, tbv aiftbv d^ XQ^Ttov xal ekatxov i^ivxvkvV'- 

SqCov dQad^i/j^etai. 
N el Si {)7tb Svo d^iidxcov bQfpxo^ (paveQdv^ 5tL xal 
ist^ aixov 6v^piJ6BtaL t& iitl xflg 6ipaCQag elQrjiiiva. 

"Akkiog. 

10 "E6X(a xvxkog^ o^ ^0X0) xivxQOV tb A^ 6ri^€L0v Se 
ixxbg i6x(o xb Z, xal iitElevxd^a &Jtb xov A iitl tb Z 
i] AZ^ xal avi/jx^(o aitb xov A 6rj^eCov xy AZ TtQbg 
dQd^&g iq)* exaxeQa x& ^iQrj 17 FA' i^ FA aQa Sl&- 
[lexQdg i6XL xov xiixkov. xal TteQLyeyQ&ipd^fo TteQl xijv 

15 AZ x^dxkog 6 ABZE^ xal iite^exix^m^av at AB^BZ^ 
ZE^ EA. ai ZBj ZE aQa itp&movxaL^ iiteLSriTteQ al 
TtQbg tolg B^ E 6rj^eCoLg el6lv dQd^aC. iitel ovv &jt6 
tLvog 6rj^eCov xov Z jtQbg xijv xov xiixkov jteQLfpiQeLav 
7tQ067te7tx<hxa6LV &xtLveg aC BZ^ZE^ xb BE ccQa [liQog 

20 bQa%"ifi6etaL xov xvxkov. l6tL Sh xb FBEA fjiiLxvxkLOv. 
xb BE aQa ikatx6v i6XLV fj^LxvxkCov, 



1. T(J] T&v m. S. zir]JNY. 4. Br] BJV V. 5. rnii' 
KvXiv$QOv Vm. 8. siQrni^iva] om. v, m. 2 Vat., :oor>^ B. 9. 

&XXo)g] BVatmv, mg. V, &XX(ag vb oc{yc6 Vat.^ Mg. X§' V. 

10. KvnXov V, O" B. Sh iyLTdg] iyiTbg Si BVat., ^aToi Si v. 

11. i5:7r6 tov A] om. Bv, m. 2 Vat. inl rb Z] om. BVat.v. 

12. r]x^a} BVat.v. crm^siov] om. BVat.v. 13. tcc f^aV^] 
om. B Vat.v. 15. v.v%Xov v, 0** BVat. iTtsSsvx^oj v, comp. 
BVat. 16. ZB] !S!B y, BZ m. 19. BE] ZE v. 20. 
forti; V. 



EUCLIDIS OPTICA. 



49 



et gz. neutrum ergo eorum secat cliilindrum. con- 
tingunt enim et dh et dg et he et gis. uidetur ergo 
sub hd ei dg hg quidem minus semicirculo. ad liunc 
autem modum et minus hemichilindro apparet. 

si duobus oculis chilindrus uideatur, manifestum, 5 
quoniam et in eo contingunt, quae in spera. 

Esto circulus, cuius sit centrum a, uero sit extra 
sitj;8?, et coniungatur a0y et trahatur a puncto a rectae 
az adfrectos in utraque gd. ea ergo gd diametrus 




est circuli. et describatur circa a0 circulus ahze^ et 10 
coniungantur ah, hz, ze, ea. itaque zh et 06 con- 
tingunt, quoniam quidem qui ad h, e puncta sunt recti. 
quoniam ergo ab aliquo puncto uidelicet ad circuli 
periferiam accidunt radii hz^ ze, ergo he pars circuli 
uidebitur. est autem ghed semicirculus. itaque he 115 

S. bg] big D. 7. uero sit] scr. punctus uero. 8. con- 
iungantur D. 12. ad] post ras. 1 litt. D. puncta] p ta, 
supra p rasi, D. recti] corr. ex recte D. 

Euclides, edd. Heiberg et Menge. YH. i^ 



50 EUCLIDIS OPTICA. 

rovto Sh rb d^sdiQrj^a ysyovs XQbg roi)g x6vovg xa 
Ttal roi)g xvXivSQOvg. i&v y&Q &jcb r&v B^ E 6rjiiBi(ov 
&%%'c}6v JCQbg dQd^itg aC TtlevQol r&v xvUvSQiov^ iq>- 
dijfovrav airmv^ xad'* o [liQog xal a[ axrtveg itQOtf" 
6 n(jtrov6tj xal a7to7iX£i6%^ifi6 Br ai ro B^E [liQog rrjg 
Htlfstog^ ^'BOQrid^ifi^ srav S\ rb BE ^iQog rov fi^LxvxlLOv» 
rb aifrb aQa ^iQog xal r&v x6v(ov d^ewQrjd^r^^Brac rb 
ikarrov. 

xd'\ 

10 Tov S^^arog red^ivrog iyyiov rov xvUvSqov eAar- 
rov iiiv i6rv rb TteQL^aiipavd^evov iitb r&v axrivcov 
rov xvXCvSqov^ Sd^ei S^ ^et^ov bQa^d^av, 

i6r(o xvltvSQog^ ox) ^d6ig lihv 6 BF xvxkog^ xiv- 
rQov S^ rb A^ Hiiiia Sh rb E^ afp^ oi ijte^evxd^co iitl 

15 rb xivrQOv ij EA^ xal JtQo67ti7trir(o6av oflcrtveg aC 
EBj EF^ xal dvT^xd^io^av ditb r&v B^ F 6rj^eL(ov TtQbg 
dQd^dg rp xvkCvSQG} al TZ^ BH, Sid Sii rd TtQdreQa 
ro HBFZ iXarr6v i6riv fnLixvhvSQCov* xal filiTterav 
iitb rov E H^^arog. ^eraxeC^d^cj tfi) ro Sft/Lta iyyiov ro G. 

20 A^yoj, ort ro JteQiXaiipavd^evov {)7tb rov S Hii^arog 
Soxet rov ZFBH [let^ov (paCve^d^ai ilarrov avrov <iv. 



4. TeQocTtiTtTovaZ m, nqoaitiTCx Bv, ut saepe. 7. %a^] 

postea add. V, om. Bv, m. 2 Vat. z&v nthvoav] VBVat.v, 
r&v Hdbvov xai r&v yivXivdQiQV Vat.*m, rov livXLvdQOv m. 2 V, 
TLvXivSQoov supra scr. Vat. m. 2. 8. Post ^Xccrtov add. 

:~ Ig^S V. 9. x-O-n Xy' V, Xa Vat.v. 10. Post prius 

Toi> ras. 1 litt. V. eyyLov^ corr. ex ^yysiov V, item lin. 19. 
14. insisvY^oiaav v. 16. icvi/jv^aj Vat., comp. B. t&v] 
corr. exToi; Vat. arnisiov Vat., sed corr. 17. TtQ&ts? m, TtQdts- 

^ BVat. 19. r6 (alt.)] to« m. 21. ZTBif] FZBH v. 



EUCLID18 OPTICA. 51 

miiior est semicirculo. hoc autein theorema factum 
est ad conos et ad chiUndros. si enim a punctis h, e 
trahantur ad punctos latera chilindrorum, contingunt 
eorum, per quam partem et radii incidentes, et in- 
dudetur hgde paxs uisus, uidebitur autem he pars 6 
semicirculi. et eadem ergo pars conorum uidebitur 
miaor. 

Prope cliiliiidrum oculo posito minus quidem est 
chilindri, quod sub radiis intercipitur, uidetur autem 
maius uideri. 10 

esto chilindros, cuius basis hg circulus, centrum 
autem a, oculus uero sit e, a quo coniungatur super 



M /^ 




centrum ea, et accidant radii eh et eg, et protrahantur 
a punctis fe, ^r ad rectos chilindro gz ei hi. per ea 
uero quae prius ihgz minus est semichilindro; et 16 
uidetur sub e oculo. transmoueatur autem oculus t 
propius. dico, quoniam, quod continetur sub t oculo, 

2. echilindros D. 3. trahatur D. punctos] scr. rectos. 
14. chilindro] chilind seg. ras. D. 16. semichilindo, sed 

corr., B. 16. e] eh B, 17. quod]^ mg. m. 1 T>. 



52 



EUCLIDIS OPTICA. 



7tQ06itL7ttEX(o6av axttv£g ccL ®K^ SA^ xal avi/ixd^io^av 
aTtb twv K^ A 0ri^€i(ov [af] TtksvQal tov xvXlvSqov 
TtQog d^d^Stg al KM^ AN. d^etoQrjd^il^staL Si^ iTth r&v 
SK^ SA axtivcov tb MKAN ^EQog tov kvUvSqov. 
6 &Uic xal 'bjtb t&v EB^ ET rb ZrBH. idti 81 tb 
ZFBH tov MKAN ^atlov* Soxel Sh Ua66ov (paivs- 
^d^ai^ iTtBiSrptBQ xal ^si^(ov ycovia rj Jt^bg tm S tfjg 
TtQbg t& E^ 

r. 

10 Kg)vov xvxlov sxovtog t^v pd6iv xal ytQbg d^d^ag 

aitfl tbv a^ova i>7tb tov svbg Hii^atog b^o^svov llat- 

tov fjiitxcoviov dipd^Tj^staL. 

i6tc3 x&vog^ o^ pd6cg ^sv 6 BF xvxXog^ xoQVfp'^ 

S^ tb A 6rj^SL0v^ a^^a Sh s6t(o tb ^, ag?' ov 7tQ06- 
16 7tL7ttst(o6av axtt- 

vsg at ^B^ ^F. 

Xal STtsl 7tQ06- 

jtS7tt(bxa6LV dxtt- 
vsg at z/r, ^B 

20 iipa7tt6iisvaL tov 
Br^ tb Br aQa 
ika666v s6tLV iiiiL- 
xvxUov Sl^ td 7tQoa7toSsSsLy^sva. ^%%^(o6av d7tb tf^g 
xoQViprjg tov x6vov trjg A S7tl td B^ F 6rjiista 7tksvQal 

25 tov x(ovov at AB^ AF. tb aQa siiL7tsQLka^^av6iLSVOv 
{)7tb twv AB^ AF sid^SL&v xal tov BF toiis(og ikatt6v 
i6tLV 'fi^LX(0VL0v^ i^tSLS^rptSQ xal tb BF ika666v i6tLV 
illlLLXvxkC(yv. ika66ov aQa ii^LX(0VL0v btpd^ifi^staL. 

1. &vifi%Q'(oeav] comp. BVat. 2. armslcc Vat. ccl'] om. 
BVat.v. 3. AN] AH By, et Vat., sed corr. 4. MKAN] 

L|i KAN BVat. (in Vat. corr.), ^y KAN v. 5. iauv v. 




EUCLIDIS OFnCA. 53 

Tiidetur eo, quod est isgiby maius apparere minus eo 
existens. accidant radii tlc^ tl, et protrahantur ab h 
et l punctis latera chilindri ad rectos hm et In. uide- 
bitur sub th et tl radiis ea quidem pars chilindri, 
quae est hmln. sed et sub eb et eg ea, quae est zgbi. 6 
est autem isgbi maior. uidetur autem minor apparere, 
quoniam maior est angulus qui ad t angulo qui ad e, 

Coni circulum habentis basim et ad rectos ei axem 
sub uno oculo uisi minus hemicono uidebitur. 

esto conus, cuius basis quidem circulus bg, uertex 10 
autem a punctus, oculus uero sit d, a quo accidant 
radii db^ dg. et quoniam accidunt radii dbj dg con- 
tingentes bg, ita bg minus semicirculo per ea, quae 
monstrata sunt. trahantur autem a uertice a coni 
super bj g latera coni ab, ag. itaque intercepta sub 15 
ab et ag rectis et bg pars minor est hemiconO; quo- 
niam et bg minor est semicirculo. minus hemicono 
uidebitur. 



1. eo (jpr.)] ea D. 12. db (alt.)] h post ras. 1 UU. B. 

17. minus] post min- ras. 1 litt. D. 

6. rov MKAN^ om. Bv, m. 2 Vat. ^Xccttov BVat.v. 7. 

tiBtiov V. 9. V] Xr V, ;Lj3' Vat.v. 10. ^xovtos] h^i B, 

lt%ovxa V. 11. ofDlova B et Vat., sed corr. m. 2. xov] 

om. BVat.v. 12. 7i{Li%(oviov] -(ovi- in ras. V. 13. n&vov v, 

%/ B. 14. TCQoamTtrita} Bv. 16. ^T] JN y. 17. ^ai 

— 19. ^B] m. 2 B. 19. JF, JB] JB, JT BVat.v. 22. 
iXatrov Vat., comp. B. 23. i]x^o) Bv, et Vat., sed corr. Tfjg] 
bis V. 24. lioivov] corr. ex y.'6%Xov m. 2 Vat. ini] inu 

Bv, et Vat., sed corr. m. 2. J5] corr. ex ^ m. 2 Vat. cri- 
[LBla] comp. post ras. 1 litt. B, m. 2 Vat. 25. TtSQLXaii^avi- 

ILSvov BVat.v. 26. TopLEoog] ro Vat., roft- v. ^Xaeaov v, 
comp. B. 27. xai] om. v. ^XaTTov Vat., comp. B. 28. 
^XaTTov Vat.v, comp. B. clqol\ iexiv ^. 



54 EUCLIDIS OFnCA. 

Aa'. 

Tov Sh Siiiiarog iyyvQv red^ivtog iv rp air^ iTCi- 
jtiSp^ iv S i6tLv ij ^&6ig rov xdivov^ ikarrov ft^v i^rai 
ro iytb r&v 8^6(oi/ i^TCSQcla^^avd^evov (liQog^ S6^€c 
5 Sh ^et^ov bQ&^d^av, 

i6rco x&vog^ 0^5 fid^Lg ft^i/ 6 AB xiixlog^ xoQvtpii 
S^ ro r 6riiistov^ Sftftc; Si i6rco rb /l^ xal elXijipd^co rb 
xivrQov rov x^ixkov rb A^ xal iTCs^siix^cs svd^sva ^ 
^A^ Tcal jtQO^' 

10 7tc7crirc36av 

&7irlvsg al /lA^ 

AB^ xal iite- 

ieiix^^co^av [af ] 

jtXevQal rov 

15 xmvov al AF^ 
rS. oixovv 
i)itb roi) z/ ^[i^arog xal r&v jdA^ AB S^foi/ i^jteQi-^ 
ka^^dverai rb ABF ^iiQog rov xdivov^ xai i6rvv 
Slarrov iiiiiXG)VLOv. [leraxeL^d^cj tf^ ro S^fia iyyLOV 

20 xal i6rc3 rb E^ xal 7tQo67tL7trirco6av dxrtveg al EZ^ EHj 
xal iyte^e^^x^co^av al Tt^vQol at ZjT, FH. jtdUv oiv 
iliTteQLka^pdveraL iitb rov E Sii^arog xal r&v EZ^ EH 
Sil^ecav rb ZFH ^iQog rov xd)vov, i6rL Si rb ZFH 
rov ABF ika66ov' Soxet S% [let^ov q^aCve^d^aLy ijteLSij 

26 ^ei^csv i6rlv fi i)Ttb ZEH yovCa rfjg ijtb A/IB ycovCag. 
q>aveQbv Si^ ort xal iytl x6vov iitb rmv Siio 6ft- 
^drcjv bQCj^ivov ^v^^il^eraL rd i^tl rrjg 6(paCQag xal 
rov xvUvSqov r&v bfioCcog bQco^ivcav 6v^^aCvovra. 




1. Xa'] Xb' V, Xy' Vat.v. 2. df] om. Bv, m. 2 Vat. 

fyywp] corr. ex fyysiov V, ut lin. 19. TS&ivtog] ts^sirccL 



EUCLIDIS OPTICA. 55 

Oculo propius posito in eodem plano, in quo est 
basis coni^ minor quidem erit^ quae sub uisibus inter- 
cipitur pars, uidetur autem maior uideri. 

esto conus, cuius basis quidem circulus ab^ uertex 
autem g punctus, oculus uero sit d^ et sumatur i 6 
centrum circuli, et coniungatur diy et accidant radii 
dttj dhy et copulentur latera coni ag^ gb. itaque sub d 
oculo et da et db uisibus includetur abg pars coni, 
et est minor hemicono. iaceat autem oculus propius 
sitque e, et accidaut radii ez et el, coniungantur 10 
latera 0g et gL rursum ergo includetur sub oculo et 
sub 60 et el uisibus gzl pars coni, quae est quam 
abg minor. uidetur autem maius apparere, quoniam 
maior est ^el angulus angulo adb, 

manifestum et in cono, quoniam sub duobus oculis 16 
uiso contingunt in spera et chilindro similiter uisis 
contingentia. 



6. coniungantur D. 16. uisis] uisus D. 



Vat., corr. m. 2; Ti&riTS v, t€ B. 4. 7CBQvXafiBav6iisvov 

BVat.v. d(J|«tf m. 8. ei^&sta] om. Bv, m. 2 Vat. 13. 

al] om. BVat.v. 20. TtgoamTtTBt B. EW] EN y, et Vat., 

sedcorr.m.2. • 21. iitstsvx^'^ B. FH] JVif BVat.v (in Vat. 
supra scr. ri). 22. EH] EN m. 23. Post S^ijjsmv del. 

t6 fiBztov cpalvscd-aL B. ZFH (utrumque)] FZH Bv, 

et Vat., sed corr. m. 2. htiv v. ds] Vat.mv, ^if V. 24. 
^XatTov Vat.v, comp. B. 25. y,slia}v] corr. ex ^st^ov m. 2 V, 
iist^ov Bv. yayvLag] om. BVat.v. 26. $i] dij v. t&v\ 

om. BVat.v. 28. dQOfiivoav v. 



56 EUCLmiS OPTICA. 

^Eav aTCo tov ^^iiatog JCQog tijv tov xcdvov ^d6iv 

7tQO67Cl7Ct(O0CV aXttVSg^ aiCO S\ tCbV 7CQO07Cl7CtOV6&V axtL-r 

v(ov xal ifpaTCto^evov aTcb t&v aq^&v svd^elai axd^&esi 

5 8l& tr^g iTCitpavsCag tov x(hvov 7CQbg tijv xoQvtpi^v avtov^ 

Sict Si twv dj[pu6G}v xal tmv dTcb tov H^iiatog TCQog 

tijV pd6LV tOV X(6vOV 7CQ067CL7CtOV6WV i^CLTCBSa SKP^rjd^j 

iTcl Sh trig 6vva(pijg avt&Vj tovts6tLV iTcl tr^g xovvfjg 

to^Tjg tcbv i7Ci7CsSov^ tb H^ua tsd^fj^ tb 6q(6^svov tov 

10 xdvov Sid Tcavtbg l6ov dcpd^T^^staL trjg S^scog iTcl TcaQ- 

akXljXov iTClTCsSov ta TCQOVTCOXSillSVCa iTCiTCsSco 'bTCaQ- 

Xoiarig. 

i6tco x&vog^ ox) pd6ig iisv 6 BF xvxkog^ xoQv^pij 
Sh ro A 6rjiiSL0Vy <i^iia Ss s6t(o tb ^, d(p* oh 7Cqo6- 

15 7CL7Ctstco6av dxtlvsg aC ^Z, ^F, xal dvrix%^o6av dTcb 
tcbv 6vva(pG)v tov Z, r TCQbg tijv xoQv^pijv tov xcivov 
tijv A ^chsvQOLl tov xcivov aC Z,A^ TA^ xal ix^s^kri^d^o 
t6 ts SLa tcbv ^Z, ZA i^CLTCsSov xal tb S^d tcbv JT^, 
FA, 7C0L7J6SL aQa tijv xoLvijv to^ijv svd^stav, ^6t(o 

20 ^ AE/i. Isyco^ ort, idv i7cl tfig AE^ [istatsd"^ tb 
S^lia^ tb t6ov tov xcivov 6(p%^ili6staL^ o6ov xal i)7cb tcbv 
jdV^ ^Z dxtLV(ov ip^sTCsto, xsL6d-c3 ydQ i7cl tfjg AEz/ 
tb (i^iia ro JS, «9?' oi) 7CQ067CL7CtstG)6av dxttvsg 7CQbg 



0) 



1. X§''] Zs' V, X$' BVat.v. 3. TtQoaTtiTttcoGLv] itQOCTtinr B, 
^QoaTtLTtritoDGav v, et Vat., corr. m. 2. 4. t&v'] corr. ex tov v. 

dcp&v] iitoKp&v m. 8. v.oivfig\ noiv&v Bv, %oi Vat. 11. 

'bytaQx B, et Vat., corr. m. 2; vTtccQxovaa v. 13. Ti&vog] 

comp. B, K&vov V. 14. TtQOGJtLTttst B. 16. dZ, JF] jr^ 
JZ BVat.v. Jr] con\ ex JNY, 18. ZA] om. Bv, 

m. 2 Vat. rj] jr BVat.v. 19. TtoLslcsL v. si^&slav] 

s^'^ BVat, sitd-ela v. 20. iTa\ corr. ex iicsl m. tf{q\ f 



EUCLIDIS OPTICA. 



57 



Si ab oculo ad basim com accidant radii, ab ac- 
cidentibus uero radiis et contingentibus a contactu 
rectae trahantur per superficiem coni ad uerticem eius, 
per protractas uero et ab oculo ad basim coni ac- 
cidentes ebipeda educantur, in contactu autem eorum, fi 
boc est in communi sectione ebipedorum, oculus pona- 
tur, uisum coni per totum aequale uidebitur uisu in 
parallelo ebipedo praesubiacenti plano existente. 

esto conus, cuius basis quidem circulus bg^ uertex 
uero a punctus, oculus uero sit dy a quo accidant 10 

radii gd et d0, 
ducaturque a con- 
tactibus 0y g 3.6. 
uerticem a coni 
latera 0a et ga, i^ 
et educatur ebi- 
pedum per dg et 
ga et d0 et ^a, 
faciet ergo com- 
munem sectionem 20 
lineam sitque ipsa 
aed. dico, quod, 
si in ad trans- 
ponatur oculus, aequale coni uidebitur, quantum et 
sub gd et dz radiis uidebatur. iaceat enim sub 2fi 
aed oculus e, a quo accidant radii ad conum. 




2. contactu] corr. ex contractu D. 
16. et — 18. ga] mg. m. 1 D. 



7. uisu] uisui D. 



supra scr. m. 1 B, rc5 v, tov Vat. iiErcct£&f[] natarsd'^ Vm. 
23. TCQoemTtrsT v. v 



58 EUCLIDIS OPTICA. 

tbv x&vov. iksii^ovtai Si^ xarcJ: titg AZ^ AF^ ijtsi,- \ 
Si^7t6Q iitl naQakki/ikov iiaTtiSov xettat tb b^^a^ xat' 
eid^sCag S% yQa^^id^g q^igovtav at 'd^^sig. sl y&Q ixtbg 
7ts6ovvtat t&v AFj AZ^ xka6%^6ovtai al Hil^SLg' SstsQ 
5 atOTtov, i6t(o6av oiv at E0^ EH, ijtsl ovv iitl Ttag- 
akkT^kov ^hv iitLJtiSov xat^ eb^^slag y^a^^d^g (piQOvtat 
a[ i^sig^ t& Sh iTtb t6(ov ytovi&v bQci^sva l6a (pai" 
vstav^ S6aL S^ av Sil^sig iitl tr^g AEA s^bd^sCag tsd^&^t 
TtaQdkkrjkoLj t6ag ycavCag jtSQiixov^v^ tb t6ov aQa tov 
10 xdvov 6(p%^6sxaL \sMsq t6ov 6Qa)6cv' ika66ov Sh tov 
xdvov bQcb6iv' &6ts xal tb ikattov dg^d^T^^stat tov 
xdvov^, 

ky\ 

ndkiv Si ys tov b^^iatog [istatsd^ivtog aitb tov 

15 tcatscvov ^stsdQOv ^ihv tov H^iiatog tsd^ivtog ^st^ov 

^hv i6tat tov x(hvov tb bQciiisvoVj S6^st Sh ika66ov 

(paCvs^d^at^ taTtSLVotiQov Sh ika66ov [ilv i6tai^ S6i,st 

Sh iLSi^ov (paCvs6%^ai, 

i6t(o xcbvog^ o^ ^A6ig ^sv 6 BF x^ixkog^ xoQV^p'^ 
20 Sh tb A 67j^stov^ xal i6to6av al TtksvQal tov xdvov 
a[ BA^ AF, iyts^svxd^co 'fi BF^ xal %Q06sx^s^kri6%'o 
t^ BF ii BH^ xal ^x^^ ^^^ '"^^^ tvx6vtog tov ® 6rj- 
[isCov t^ AB ytaQdkkrjkog fi ®K, kiya^ oti ^st^ov 
lihv i6tai^ ika66ov Sh 6(pd"il6stai tov xdvov tb b^d- 
26 ^svov tov Siiiiatog tsd^ivtog iitl tov S 6rj(isCov ^TtSQ 



3. q)aLQOVtai v. i%T6g] v supra scr. m. 2 V, ivr6g 

BVat.^m. 4. AT, AZ] corr. ex AFZ m. 2 V, mut. in ATZ 
m. 2 Vat., AF, ZA Vat.\ 'nXaLa&ricovraL v. 8. Post &v 

del. «i m. rfjg] roH Vat., rijv v. 9. nBQLiB Vat., TtSQisxst' v. 

10. elheeQ'] mut. in SnsQ m. 2 Vat., SitSQ Vm. Uarxov 

Vatv. dt] &Qa V. 13. Xy'] Xf V, Xs' Vat.v. 14. ^era- 



EUCLIDIS OPTICA. 59 

procedunt autem secTmdum az et ag, quoniam in 
parallelo ebipedo positus est oculus^ secundum autem 
rectas lineas finitur uisus. si enim extra cadunt ag, az^ 
franguntur uisus^ quod locum non habet. sint ergo 
ety ei, quoniam ergo in parallelo ebipedo secundum 5 
rectas lineas feruntur uisus, quae uero sub aequalibus 
angulis uisa sunt^ aequalia apparent^ quotcunque autem 
uisus in aed recta ponuntur paralleli, aequales angulos 
continent; aequale ergo coni uidebitur. et si aequale 
uident, minus uero coni uident. quare et minus uide- 10 
bitur coni. 

Rursum autem oculo transportato ab humili ele- 
uatiorique oculo posito maius quidem erit de cono 
uisum, uidetur autem minus apparere, humiliori uero 
minus quidem erit, uidetur autem maius apparere. 15 

esto conus, cuius basis quidem hg circulus, uertex 
autem a punctus, sintque latera coni ba^ ag, et con- 
iungatur bg, et adiciatur rectae hg recta hi^ et traha- 
tur per punctum contingens t rectae ah parallela tJc. 
dico, quod maius quidem erit, minus uero uidebitur 20 
de cono uisum oculo posito super t punctum quam 

7. quocunque D, corr. m. 1. 8. paralell seg. ras. 1 Utt. D. 
21. de] om. D in extr. pag. 

ts&^ivtog] yLccrccred^ivtos lu. 15. usrsmQOTEQOv BVat.v. 16. 

^Xattov Vat.vm, comp. B. 17. twjtsivo Vat.v (in Vat. corr. 
m. 2). Harrov Vat.v, comp. B. 21. AF'] A ixL ras. m. 

iitsiBvx^'" Bv. 22. 7{\ 6 V. tov (pr.)] om. Bv, m. 2 Vat. 

24. ^locttov Vat.v, comp. B. 25. tsd^ivtog] ts^^ Vat., corr. 
m. 2; tid"ritoci v. 



60 



EUCLIDIS OPTICA. 



iTtl tov K. ijCEievxd^iXi^av av AK^ AS^ xal 7tQ06- 
expBp^^d^io ii AS iTcl rb H^ rj S^ AK iud xh A. 
oixovv i%l xov H xal rov A red^ivrog rov Sii^arog 
avi6a rS: bQci^eva rov xdvov dfpd^ij^eraLj xal ^ev^ov 

5 [ihv i^rau ro iCQog rp if, iXa66ov S\ Sv iiet^ov dipd^- 
6eraL rb TCQog ra A. t6ov S\ ro JtQog r& H rp TtQog 
ro @5 ro Sl TtQbg r» A rp TtQog rtp K^ &g iv r& 
TtQb rovrov iSei%%"Yi. rov aQa Sftftarog ^tQbg r& & 
red^ivrog iiet^ov i6rai, ro bgio^evov rov xcovov i^TteQ 

10 JtQbg rc5 K^ S6^ei Sl eka66ov elvai. 

kS\ 

'Ed^v xvxXov TtQbg dQd^d^g aTtb rov xivrQOv ava6rad^ 
rc3 rov x^^xXov iitiTtiSm ^d^eta^ iitl S\ ra^&trig rb Siiiia 
red^flj ^^ Sid^erQoc aC iv rcf rov xvxXov iTtiTtiSp Stay^' 
15 inevaL 7ta6aL l'6aL ipav7}6ovrai. 

i6ra) xvxkog^ o-S xivrQov ro A 6i]^etov^ xaX iat' 
avroi) avilijipGi rig TtQog dQd^ag 7] AB r^ rov xiixXov 




ijtvTtiSip^ ifp^ '?jg b^iia xeC^d^co ro B. Xiyo^ ort al Sid- 
(lerQOL t6aL g)avi/j6ovraL. i6rco6av Svo Sid^erQOL av 

1. ToC] t6 m. 2. H] e corr. V. r6 (alt.)] corr. ex td 
^. XBU 2 Vat. 3. TS&ivrog] zbQ^bixoli Vat., corr. m. 2. 5. IsGtai] 
^aa, eodd. x6 — 6. laov ds] m. 2 Vat. 5. rw] r6 Bv. 



EUCLIDIS OPTICA. 



61 



super Tc, coniungantur aifc, at, et eiciatur at super i 

et ak super l. nunc ergo super l Qi i posito in- 

aequalia^ quae 
uisa sunt cono, 
uidebuntur , et 5 
maius eo quod 
ad i id quod ad ty 
eo uero quod 
ad l id quod 
ad hy sicut in 10 
praemissa de- 
monstratum est. 

oculo ergo ad t posito maius erit, quod uidetur cono, 

quam ad k, uidetur autem minus esse. 




Si a centro circuli ad rectos circuli ebipedo eriga- 16 
tur recta, atque in eo oculus ponatur, diametri in 
circuK ebipedo ductae omnes aequales apparebunt. 

esto circuli centrum a punctus, et ab eo trahatur 
perpendicularis circuli ebipedo, in qua iaceat oculus 6. 
dico, quod diametri aequales apparebunt. sint duae 20 
diametri dg et ez, et coniungantur hg, he, hd, bz. 

1. eciatur D. 2. posito] scr. posito oculo. 4. cono] 
8cr. de cono, ut Un. 13. 18. punctus] -us e corr. D. 20. 
duae] duo D. 



^Xccacov — 6. r© H] om. Bv. 5. ^Xattov Vat. 6. to (pr.)] 
Tc5 m. 1 V, tov Vat.^m, V m. 2. rc5 (tert.)] t6 Bv, et Vat., 
sed corr. m. 2. 7. rcJ] tov e corr. m. rc5(terfc.)] t6 Bv, et Vat., 

sed corr. m. 2. 9. t8&ivtog] ti&v BVat., rs®^ v. iJTtSQ] 
nsQ post ras. 1 litt. Vat. 10. ^Xccttov Vat.v, comp. B. 11. 
Xd'] Xr\' V, Xz' Vat.v. 13. d«] m. 2 Vat. 14. tov] om. 

codd. 17. &vCxd'(a v; in hoc uocab. des. B. yLvyLXov] corr. 
ex 'nivtQov m. 2 Vat. 18. ^g] ^ Vm, al\ ol m. 



62 



EUCLIDIS OPTICA. 



r^, EZ^ xal inalaiix^(o6av at BF^ BE^ B^d^ BZ. 
aTCal ohv t6ri i^tlv f] ZA tfi AF^ xocvii dh fj AB^ 
xal dQd^al aC yonvCat^ pd6Lg &Qa fj ZB pd6aL tfj BF 
t6ri i6TLv^ Kal al TCaQl tag Pcc6aLg ycovcaL. t6ri &Qa 'fj 
6 {)7cb t&v ZB^ BA tfi ino t&v AB^ BF. d^oic^g xal 
^ imh EBA tfi {}7cb AB^. i^ &Qa imb t&v FB^ B^ 
i'6rj i6tl tfl iycb t&v EB^ BZ. toc d' 'bnb t&v t6(ov 
ycjvL&v bQdfiava t6a fpaCvataL. t6ri aga ii F^ tfj EZ. 
x&v fj aTcb tov xavtQOv dx^^at^a fiii TCQbg d^d^dg ^ 

10 t& iTCLTCaStp^ t6ri da ri ty ix tov xivtQOv^ aC SLdyuatQOi 
7Ca6aL t6aL (pavil^ovtaL, 

i6t(o x^ixXog 6 ABF^^ xal ^%^(o6av alg aitbv di5o 
dLd^atQOL at AB^ P!^, xal i6tc3 fi djcb tov E 6riiiaL(yv 
dvayo^avri^ i(p^ fjg tb &iiiia 

15 xattaL tb Z, ft^ TCQbg dQd^dg^ 
dXXd t6ri axd6tri t&v ix tov 
xavtQ(yv fj ZE^ xal ina- 
t^aiijip^Gi^av dxttvag at ZA^ 
ZT^ ZB^ Z^. ijcal ohv t6ri 

20 i6tlv fi BE ty EZ^ dXXd 
xal fj EA t6r} i6tl tfi EZ^ 
at tQalg &Qa at EZ^ EA^ EB t6aL ai6Cv. tb &Qa 
iv tp dLd t&v AB^ EZ imniSc} naqX tijv AB SLd- 
fiatQOv fifiLXVKXLOv yQa(p6^avov ikav6ataL S^d tov Z. 

26 dQd"^ &Qa fi fjTcb t&v AZ^ ZB. b^oCcog xal fj i^b 




2. Tjl T^s Vat.v. AF^m ras. V. %0Lvii 8\ ii AB'\ 

supra scr. V; del. est: tari ^<"^of* >ta^ i] ya xfj aB. 3. %alyo:iviai 
dg&aL Vat.v. ai] om. Vm. ZB] BZ Vat.v. BF] in 

ras. V, AF m. 4. iatLv tari Vat.vm. 5. x&v (utnimque)] 
om. Vat.Vat.Wm. BA'] A Vat.»m, B del. VVat. AB"] B 
del. V. Bri B del. Vat., F Vat.^m, F in ras. V. 6. EBAli 
^^B r, et vat., sed corr. tfj 'bno] postea add. V. ABJj 



EUCLIDIS OPTICA. 63 

qnomam ergo aequalis ea ei^ quae est ag^ communis 
uero ahy et anguli recti^ basis igitur hg basi hg est 
aequalis^ et qui circa bases anguli aequales. ergo qui 
sub ssh^ ha angulus angulo qui sub ahy hg. similiter 
et eha angulus angulo ahd. et qui ergo sub ghy hd 5 
aequalis est angulo qui sub eh, hz, sed sub aequali- 
bus angulis uisa aequalia apparent. aequalis ergo recta 
gd rectae ez. 

et si quae a centro ducta non perpendicularis 
fuerit ebipedo, aequalis uero ei quae a centro, dia- 10 
metri omnes aequales apparent. 

esto circulus ahgd, et trahantur in eo duae dia- 
metri ah^ gd, et sit recta ab e puncto ducta, in qua 
oculus positus est 0, non perpendicularis^ sed aequalis 
unicuique earum, quae a centro, uidelicet e0, et ducan- 15 
tur radii 0a, zg, zhy zd, quoniam ergo aequalis est 
he rectae eg^ sed et ea rectae ez aequalis; hae tres 
ergo aey ez et eh aequales sunt. in eo ergo per ah 
et ez ebipedo descriptus semicirculus circa ah dia- 
metrum ueniet per z punctum. rectus ergo qui sub 20 
az et zh angulus. similiter et qui sub gz^ zd rectus 



1. comunis D. 



A postea add., B/^ e corr. V. 17] hl v. z&v\ om. Vat.^m. 

rBl B eras. V. B/il B del.Vat., d Vat.^m. 7. Twi;(pr.)] 
om. Vat.^m. EB] B del. V. BZ] B del. Vat., Z Vat.^m. 

r&v (alt.)] om. Vat. Vat. Wm. 9. Xd'' V, Zf m. 2 Vat. n&v] 
idv m. 13. if] om. v. 14. Scvaydnsvov v. 17. ZE] £Z 
Vat.vm. 21. l'67i i6tl]^om. Vat.v. EZ] EZ l'6ri Vat.v. 

22. EZ, EA] AE,EZ Vat.v. si^i Vat.v. 23. nsQi — 

24. iX€^66tocL] yQaq)6iievov ijiiLHvyiXtov (corr. ex -l(p m. 2 Vat.) 
negl tiiv AB duHiistQov tj^sl %cci Vat.v. 25. t&v] om. Vat.^m. 

AZy ZB] AZB Vat.^m. ZB] Z eras. V. 



64 



EUCLIDIS OPTICA. 




r&v rZ^ Z^ i6tLv dQd^ii. af dh dgd^al t6av^ ric 81 

vTcb t6(ov y(ovi&v 6Q(hfi6va t6a (paCvetai, t^rj aqa (pa- 

vrj6staL xal fi AB xfi F^. 

aXXo: Sii fi AZ [iilts i'6ri l6t(X) tf^ ix tov xdvtQOv 
5 fin^ts TtQog dQd^ctg tS tov xnixXov ijciTCsSco^ t6ag Sh 

y(ovCag TCoisCtco tdcg 'bno ^AZ^ ZAF xal to^g iTcb 

EAZ^ ZAB. Xsy(o^ 5tc xal ovteog al SLccfistQoc l6aL 

(pavri6ovtaL aLTCOL- 

ov6aL to^g l'6ag y(o- ^ 

10 vCag. 

iicsl ycLQ l'6aL 

si6lv al [ikv FA^ 

AZtalgZA^A/i^ 

al S\ BA, AZ 
15 talg ZA^ AE^ xal 

aC ycDvCaL t6aL^ 

^&6Lg ccQa ii ^Z Pcc6sl ty ZF t6ri i6tCv' &6ts xal ij {)7cb 

jdZA l6ri tfi ^%b AZr. bfioCcog Sij SsC^ofisv^ otL xal 

ri {)7cb EZA l'6rj i6tl tfj ^Tcb AZB. olri aga rj iTcb 
20 ^ZB L'6rj i6tl tfj iTcb EZT. &6t6 xal al JB^ ET 

SLcc^stQOL l'6aL (pav7]6ovtaL. 

Xs\ 

^Eccv S\ fj ccTcb roi) H^fiatog TCQbg tb xsvtQov tov 

xvxlov 7CQ063cC7Ctov6a in/jts TCQbg dQd^ccg y tm imTCsSfo 

25 tov x^Axkov {Li^ts tfi ix tov xsvtQov t6ri ii7]t6 t6ag 

ycovCag 7C6QLi%ov6a^ al SLcc^6tQ0L avL60L (pavifj6ovtaL^ 

TCQbg ctg 7C0L6l avC6ovg ycovCag. 

1. t&v] om. Yat.^m. ZJ] Z del. V. dQ&rj iaxiv 

Vat.v. 4. ft' V, Xri' m. 2 Vat. 17] %al i] Vat.v. 6. %al 
rag i>n6'] in ras. V, x^g vn6 om. v, m. 2 Vat. 7. ovrcoff] 



EUCLIDIS OPTICA. 65 

est. omnes uero recti aequales. sub aequalibus autem 
angulis uisa aequalia apparent. aequalis ergo apparet 
ah ei quae est gd. 

sed quod a0 nec aequalis ei sit quae a centro nec 
perpendicularis circuli ebipedo, aequales autem angulos 6 
faciat da^ et zag et eaz et ^ab, dico, quod diametri 
aequales apparebunt facientes aequales angulos. 

quoniam enim aequalis est angulus quidem gais angulo 
0ab angulusque hd0 angulo ^ae, basis ergo ^b basi 0g 
est aequalis. quare et i^a aequalis angulo a0g, 10 
similiter autem demonstrabimus, quoniam et e^a angu- 
lus angulo a^d est aequalis. totus ergo dsi toti e^g 
est aequalis. quare dh et eg diametri aequales ap- 
parebunt. 

Si recta ab oculo centro circuli incidens ne per- 15 
pendicularis faerit ebipedo circuli neque ei quae e 
centro aequalis neque aequales angulos continens, dia- 
metri inaequales apparebunt, ad quas facit inaequales 

4. ei] mg. m. 1 D. 11. demonstrabibiinus D. 15. ne] 
8cr. neque. 

om. V, m. 2 Vat. 8. al TtoLo^aai, tdg] yial itoir^aovGiv codd. 

11. iW — 16. t6Cii\ hri iarlv ij ilsv FAZ tfj ZAJ, fj dl 
BAZ Tjj ZAE ycavicc lari Yat.v; in Yat. m. 2 scripturam nostram 
restituit (12. FA'] ' dA. 14. ds] om. 15. ZAE. 16. ai\ om.). 

12. sIgIv al] in ras. Y. rA] dA Yat.^m, in ras. Y. 13. 
ZA] m. 2, Z m. 1 Y. 14. a\ ^e] m. 2, ^e.m. 1 Y. BA\ 
m. 2, B m. 1 Y. 15. rarg] in ras. Y. Z^, AE] ZAE\. 

16. wl\ om. YYat.^m. 17. ^Z] ^B Yat.v. icxiv hri 

Yat.v. 19.^EZ^] Z e corr. Yat. fari iarl'] om. Yat.v. 

AZB iativ iari Vat.v. ccQa] om. v. 20. iari ^<^^0 ^^V 

Yat.v. EZr iativ iar\ Yat.v. 22. W] {lcc' Y, Xf v, Yat. 
m. 1, X^' Yat. m. 2. 23. ds] om. ^, m. ^ '^^X.. 

EnclideBt edd. Heiberg et Menge. "SVL. ^ 



66 



EUCLIDIS OPTICA. 




s6ta) xvxXog 6 ABF^^ xal Vix^^QOfSav tft5o dLd^stQOL 
al AF^ B^ tiiLV0v6av aXXr^Xag TCQbg dQd^ocg xat& to E 
6rjfietov^ xal ij ayth tov E (SrnLeCov avayofievri^ i<p^ ^^ 
tb a^fia xeltat^ fj ZE 
6 ^ijr^ ytQbg dgd^ag SetcD 
t^ incTceSa) (lilte t6rj tfj 
ix tov xivtQOv {Lifite i!6ag 
ycaviag 7CeQii%ov6a (leta 
t&v A r^ ^B. Xiyca^ 
10 otL avL0OL 6(pd'7]0ovtaL aC 
AF^ ^B SLafietQOL. iTCe- 

ZA^ Z^, ZB. ^tOL oiv iieL^ODv i6tlv fj EZ trjg ix 
tov xivtQov t} iXd(S6(DV, Sl& tavta Sij V(tOL fieL^cav 
16 i0tlv rj i)jcb ^Z, ZB trjg hxb FZ^ ZA rl fj 'bicb 
t&v rZ^ ZA trig ijcb ^Z^ ZB^ hg ei/fig SeL^o^iev» 
avL60L aQa al SLa^ietQOL 6(pd"tJ6ovtaL, 

Afjfifia, 

"E6t(o x\)xXog^ ox) xivtQOV e6tc3 tb A 6rj(ietov^ bii^ia 
20 S^ tb B^ d(p' ov iitl tbv x^ixkov xdd^etog dyofiivrj ft^ 
TCLmitG) ijcl tb xivtQOv tb A^ dXX^ ixtdg^ xal e6tG) ij 
BF^ xal iTce^evx^co dicb tov A i%l tb F rj AF xal 
djcb tov A iTcl tb B rj AB. Xiyca^ otL 7Ca6cbv t&v 
ycovL&v t&v TCeQLexofiivcsv 'bjcb t&v Slcc tov A SLayo- 
25 fiivc3v e^dd^eL&v xal ^olov6&v Tcgbg tfj AB eid^eCcc 
ycDvCav iXaxC^trj i6tlv ij vTcb t&v FA^ AB. ^x^^ 



2. xB{LvovGLv Vat., sed corr. 
ras. V. 4. ZEl EZ Yat.v. 



3. iLvayo\Livri\ prius a in 
11. ^B] B^ Vat.v. 13. 



Zd^ ZBJ ZB, Zd Vat.v. ^LBltov v. 14. iXdxt(ov v. ravra] 
«Jr aifrdYm. 16. JZ, ZB] JZB Vat.^m, e corr. V. TZ, 



EUCLIDIS OPTICA. 67 

esto circulus dbgd et duae diametri ag^ bd se in- 
uicem ad rectos angulos secantes ad punctum e, et 
ab e puncto ducta, in qua oculus positus est, e0 neque 
perpendicularis sit ebipedo neque aequalis ei quae a 
centro neque aequales augulos continens cum dg, ah, i 
dico, quod inaequales apparebunt ag, bd diametri. 
coniungantur enim zg, za, zb, zd. aut igitur maior 
. est eis ea quae e centro uel minor. propter haec uero 
uel minor angulus qui sub dZy zb eo qui sub gz, za 
uel qui sub gZy za eo qui sub dz, 0b, sicut deinceps 10 
demonstrabimus. inaequales igitur diametri uidebuntur. 

Esto circulus, cuius centrum sit a punctus, oculus 
autem 6, a quo super circulum cathetus ducta non 
cadat super centrum, sed extra, et sit bg. et con- 
iungatur a puncto a super g recta ag, adhuc autem 16 
et ab a super b. dico, quod omnium contentorum 
angulorum sub ductis per a punctum et facientium 
angulos ad a6 rectarum angulus minimus est qui sub 
ga, ab. trahatur enim per a punctum dae. dico, 



6. continens] contingens D. Q. bd] corr. ex hg m. 1 D. 
9. eo — 10. zli] mg. m. 1 B. 



ZA] rZA Vat.^m, e corr. V. rj (alt.)] om. Vat.^m. 16. 
rz, ZA] rZA Vat.^m. M^M t&v Vat.v. JZ, ZB] 
JZB Vat.^m. 18. ^r^ftfia] om. Vat.vm, ;ij3' V (j3 e corr. m. 1), 
fi' m. 2 Vat. 19. Bato) (alt.)] om. m. 22. A] in ras. V. 

r] in ras. V, V sif&sla Vat.v. AF] seq. ras. 1 litt. V, 

AB Y. Dein add. hi $i Vat.v. 23. ri AB] om. codd. 24. 
yavL&v t&v] om. Vat.v. ^bnd] ycavi&v 'bTcd Vat.v. ^tayo- 
lUvGiv] ducyisniivav v. 26. tjj] t6 v. 26. ycaviocs m. t&v] 
om. m. FA, AB] FAB m. 



68 EUCLIDIS OPTICA. 

Sia tav A si&eia -^ ^AE. Xiym, ozt -^ vxb FAB 
rfjg isro EAB Hdeemv ieriv. ^xd'^ y^P '^** ''^''* ^ 
izl T^v jdE xif^d-iro? iv zm ixixiSm fj FZ, xal ixs- 
^svx^i' ^ -8Z. xal ^ 
B BZ aQa ixl tiiv JE 
xd&ezdg ietiv. ixBl 
o5v 6q»^ 'fi vab rZA, 
^ imb ATZ Slqk iXda- 
6<av 6Q9^g. tijv Sl 

10 fiiilova ytaviav ^ ^si- 
tfav ■scktvQKi. iittyreivsi. 
ftsi^av Spa i} AF T^g 
AZ. ilX' ^ vxb tS>v 
AT, ra xa\ ii vnb 

15 t&v BZ, ZA 6Q»aC 

sieiv mets Bielv aC PB, BZ avteoi. xal ij i>ii6 t&v 
ZA^ AB &Qa f^s iiab t&v VA, AB ieti ftsi^oiv. bfioimg 
Sii Ssiy&^estai xal «aO&v r&v ytavimv tav xsQttjOfiivmv 
iiTtb r&v Sta ttotj A Siayoftivmv f^&sietv xal noioveStv 

20 atfbg f^ AB si&sia ymviav iXa^ietri ij i)xb tav FA, AB. 

xal <paveQ6v, oti, i^v Staj^^^ ttg xal oWij sv^^Bta 

dti Toij A hg if A® noQQthtsQov or>ea tijg AF i^ntQ 

ij AZ, fisitmv ietai ■^ i)xb BA& tiig ixb BAZ. ax&si- 

erig Y&Q TcdXtv xa^itov iicl f^ A® r^s FK ixt- 

26 tevx^^staa ij BK xd^stog icrat 6ftoimg ixl ti^v A@. 
xal ixtl (isi^av ^ AA tijs AK {6q&^v y^Q iiaotsivsi 
iri)v i)xb AKA)^ aoXX^ aQa ^ AZ t^g AK fitit/ov 




1. St&\ in ras. V. io6] corr. ex tiS au 2 T. ev9tia\ 

m. 2 Tat.; mjftiioo t, Tat, m. 1. 2. HAeaiav] IXaxlatn Tm, 
UaMnv V. 3. JE] AE v (in Vat. d ^ A difflcilliiiie digno- 
SCDntuT). T^] x^ inoxeiitevoi Tat.T. 4. BZ] ZB Tat.T. 



EUCLIDIS OPTICA. 69 

quod angulus gah angulo eah minor est. trahatur 
enim a pimcto g super ae cathetus in subiacenti ebi- 
pedo, et coniungatur zh, zh ergo super ae cathetus 
est. quoniam ergo rectus gza, angulus ergo agz 
minor recto. maiori angulo maius latus autem sub- 5 
tenditur. maior ergo ag quam az, uerum anguli qui 
sub ag et gh et anguli qui sub hz et za recti sunt, 
et ghy hz inaequales. ergo qui sub za, ah eo qui 
sub ga, ah maior est. similiter autem demonstrabitur 
et onmium angulorum contentorum sub ductis per a 10 
rectis et facientibus ad a& rectam* angulum minimus 
est gah, 

et manifestum, quoniam, si demonstretur et alia 
recta per a ut at remotior existens ab ag quam az^ 
maior erit hat quam haz, tracta enim rursum catheto 15 
gTc super at coniungatur hh cathetus similiter super at. 
et quoniam maior est alg quam afc; recto enim sub- 
tenditur ag\ multo ergo az quam alc maior est. et • 



3. coniugatur D. ergo] in ras. D. 6. autem] mg. 

m. 1 B. 7. anguli] anguli B. 16. coniungantur D. 17. 
alg] scr. al. 18. ag'} scr. aJcl. 

5. BZ] Z in ras. V. JE] AE Yat.v. 8. ATZ] corr. 

ex A Y, AZ supra scr. B Vat.\ BAZ m. iXdaaova v, iXdt' 

tcav Vat. 11. ^Ttotdv^ v. 13. AZ] BZ m. 17] al 

Vat.^m. M — 16. ZA] vnb AFB, BZA Vat.»m. 14. 

rs] r del. VVat. 15. BZ] Z del. Vat. ZA] Z del. V. 

16. sioLv] stcL Vat.v. &6ts siaiv] om. VVat.*m, xa/ Vat.v. 

BZ] BZ &QOC Vat.*m. t&v ZA, AB] ZAB m. 17. t&v 
rA, AB] FAB m. ietiv v. 20. AB(^t.)]_A m. yooviag m. 

t&v FA, AB] FAB m. 21. (pavsQ^v] 9"?' Vat.^ q^aaiv m. 

diaxQ^i] dstx&^ Vat.Vat.*mv (V?). 22. itoQ&tsQov Vat.v. 

23. ^sl^ov V. 24. r^ff TK inl tijv A® Vat.v. ini' 

isvx^fjca V. 26. AA] corr. ex AJ V. 6q^i\ v^ CQm^.VaA.. 



70 EUCLIDIS OPTICA. 

i6t£v, xai slcfiv dQd^al al 'bTCo BZA^ BKA. iXcc66cjv 
^ihv &Qa fi BZ ri}g BK diic th t6a slvaL td ts axb 
t&v BZ^ ZA xal ta &7cb t&v BK^ KA tai icitb tfjg 
BA xal (JAAifAofcg, iisi^cav Sh tcAXlv ij imb BAK f^g 
5 {mb BAZ. Tcadcbv S% t&v Tcgbg ty BA yLvofiivcjv 
ya)vcG)v i)7cb t&v Stct tov A SLayoybivcDV iLsyCdtri i6tlv 
ij {}7cb BAH iK^Xrid^sC^rig tfig FA iicl tb H^ i%sl Tcal 
7Ca0G)v iXdttcov rj ijcb BAF, l'6ac Sh yCvovtai at l6ov 
a7ci%ov6aL itp^ sxdtSQa tfjg MA tfjg f^v iXa%C6tYiv 

10 ycovCav 7CSQtsxov6r}g ^istd tf^g BA. xsC^d^co y&Q tfi EM 
l6ri fi MN^ Tcal inst,svx%'G:)6av at EM^ MN^ EF^ FN^ 
BE^ BN^ AN. i%sl oiv i'6ri i6tlv ^ MN tf} ME^ 
xoLvii Si {j MjT, xal ycovCag t6ag 7CSQiixov6LV^ t6rj aga 
xal fi ET tfj FN. xolv^ Sh ocal 7CQbg dQd^&g ^ FB. 

15 t6ri ccQa xal ij EB tfj BN. dUa xal ^ EA tfj AN' 
xal xoLvfi ij AB. xal ycovCa aqa fj i^cb EAB ty imb 
NAB t6rj i6tCv. 

''E6tco xiixkog 6 ABF^^ ov xivtQOV tb Z^ iv S 

svd^staL ^x^^^^'^ ^^^ ^^^ -^9 -S? ^? ^ ti^vov6aL dXXij-- 

20 Xag 7CQbg dgd^dg^ bfifia Sh s6ta) tb E^ dtp^ oh ij i7cl tb 

xivtQOv i7CL^svyvv^ivrj 7CQbg d^d^dg tfj FA^ 7CQbg Sh 



1. iariv'] iaxi Yat.vm. BZA] ZBA Vat.v. Uccttov v, 
iXdttojv Yat. 3. BZ] ZB Yat.^in. tijg] t&v Vat. 4. 

(istiov T. ds] corr. in &qoc V, &q(x Vat.Vat.*mv. 6. duc- 
yoiisvcav] dttc- in ras. V. 7. iTCsL] yial i^tsi Vat.v. 11. MN] 
corr. ex MF Vat. EF] om. m. 13. nsQisxovai v. 15. 
BN] BH m. 16. rj] r^g Vat. 17. NAB] e corr. V. 

18. fty' V, fta' m. 2 Vat. TistQov v. 19. ijx&aaccv] ijx^ 
Vat. &XXi/jXai,s m. 20. tb yisvtQov] tov %svtQOv m. 21. rj] 
cojT. ex ^ Y. 



EUCLIDIS OPTICA. 71 

sunt recti hza et iTca [cum hza et hTca anguli 

trianguUs sunt recti, tunc quadratum gh et za ualent 
quadratum ha. similiter quadrata lch, ha ualent qua- 
dratum ha per elementa. et non igitur et inter se 
sunt aequalia^ cum ualeant idem. sed quadratum za l 
maius est quadrato ha, quia za maior, sicut probatum 
est ergo quadratum hh est maius quadrato zh, ergo 
hh maior zh']. minor ergo hz quam hh linea propter 
aequalia esse quae ab hz, za et ab hh, ha ei quae 
ab 6a et ad inuicem. maior ergo rursum hah quam ic 
haz angulus [quoniam uero hag et hai anguli ualent 
duos rectos, similiter haz et had anguli ualent duos 
rectos. igitur ualent inter se. cum igitur hag sit 
minor haz, et hai erit maior had et sic de aliis]. 
omnium uero ad ha factorum angulorum sub ductis 15 
per a maximus est hai educta ga super i, et quo- 
niam etiam omnium minor est hag. aequales uero 
fiunt aequaliter distantes ex utraque parte lineae ma 
minimum gah angulum continentis. iaceat enim rectae 
em aequalis mn, et coniungantur em, m7i, eg, gn, he, 20 
hn, an, ae. quoniam ergo aequalis est mn ei quae 
esi me, communis uero mg, et aequales angulos con- 
tinent, aequalis ergo eg recta rectae gn. communis 
perpendicularis gh, aequalis ergo et eh ei quae est hn. 
sed et ea ei quae est an, communis ergo ah. et 25 
angulus ergo aeh angulo nah est aequalis. 

Esto circulus aghd, cuius centrum z, in quo rectae 
trahantur per centrum ah, gd se ad inuicem perpen- 

1. Post anguli litt. quaedam dubiae D. 4. non] no D. 

13. inter se] bis D, sed corr. 

i 



66 



EUCLIDIS OPTICA. 




e6tc3 TcvxXog 6 ABF^^ xal ^xd^co^av dt5o Slu^btqol 
ui AF^ BA tiyLV0v6aL allif^lag ytQog dQd^ctg xatA rb E 
6rjii6toVj xal i} aTCo tov E 6ri^aCov avayofievri^ itp^ 'fjg 
tb (ififia xBvtat^ fj ZE 
6 fiTite TtQog oQd^o^g i6tc3 
t& iTCCTtiSa) (lilts t6ri tfj 
ix tov TcivtQOv (iT^ts tdag 
ycDvCag mQii%ov6a ftaro: 
t&v AFj ^B. Xiycn^ 
10 ZtL avv60L 6(p%"/}6ovtaL at 

AFj ^B SLaflBtQOL. iTCB' 

^Biixd^co^av yccQ af ZjT, 
ZA^ Z^, ZB. ^tOL oiv (iBL^cav i6tlv ^ EZ trjg ix 
tov TcivtQOv '?) iXd66a)v. SlS: tavta di) i^tOL (lei^ayv 
15 i6tlv fi ijtb ^Z, ZB f^g ijtb FZ^ ZA rj fj iTth 
tmv rZ^ ZA tflg {)7tb jdZ^ ZB^ hg ii/fig Sbl%0(ibv. 
avL60L aQa al SLd(iBtQOL dtpd^i^^ovtaL. 

Afi(i(ia. 

"E6tG) KiixXog^ oi xivtQOv B6ta) tb A 6rj(iBL0v^ &(i(ia 
20 Sh tb B^ d(p^ 0-5 ijtl tbv tcvxXov xdd^Btog dyo(iivrj (lif 
TtLTttitG) iTtl tb ocivtQOv tb Aj dXX^ ixtdg^ xal i6ra) ij 
BF^ xal iTtB^Bvx^a) d%b tov A i%l tb F fj AF xal 
djtb roi) A i%l tb B fj AB. Xiyo)^ StL 7ta6a)v t&v 
ycDVLcbv t&v JtBQLBxo(iiva)v 'bnb t&v S^d tov A SLayo- 
25 (livcov Bid^BL&v xal 7toLov6&v TtQbg tfj AB Bid^BCa 
ycDviav iXaxL^trj i6tlv rj ijtb t&v FA^ AB. Vjx%^o 



2. rsy,vovaiv Yat., sed corr. 3. &vayofisvrj] prius a in 

ras. V. 4. ZE] EZ Yat.v. 11. JB] BJ Yat.v. 13. 

ZJ, ZB] ZB, ZJ Vat.v. iisiSov v. 14. iXdrtoDV v. raihra] 

rd a^dtd Ym. 15. JZ,ZB] dZB Yat.^m, e corr. V. TZ, 



EUCLIDIS OPTICA. 67 

esto circulus dbgd et duae diametri ag^ bd se in- 
xiicem ad rectos angulos secantes ad puuctum e, et 
ab 6 puBcto ducta, in qua oculus positus est, ez neque 
perpendicularis sit ebipedo neque aequalis ei quae a 
centro neque aequales augulos continens cum dg, ab. 5 
dico, quod inaequales apparebunt ag, bd diametri. 
coniungantur enim isg, za^ zby zd, aut igitur maior 
. est ez ea quae e centro uel minor. propter haec uero 
uel minor angulus qui sub dZy zb eo qui sub gz, za 
uel qui sub gZy za eo qui sub dZy zby sicut deinceps 10 
demonstrabimus. inaequales igitur diametri uidebuntur. 

Esto circulus, cuius centrum sit a punctus, oculus 
autem 6, a quo super circulum cathetus ducta non 
cadat super centrum, sed extra, et sit bg. et con- 
iungatur a puncto a super g recta ag^ adhuc autem 15 
et ab a super 6. dico, quod omnium contentorum 
angulorum sub ductis per a punctum et facientium 
angulos ad a6 rectarum angulus minimus est qui sub 
gay ab, trahatur enim per a punctum dae, dico, 



6. continens] contingens D. 6. hd'] corr. ex hg m. 1 D. 
9. eo — 10. zh] mg. m. 1 J). 



ZA] rZA Vat.^m, e corr. V. 7] (alt.)] om. Vat.^m. 16. 
PZ, ZA] rZA Vat.^m. {fytd]^ r&v Vat.v. JZ, ZB] 
JZB Vat.^m. 18. Afjinioc] om. Vat.vm, ;ij3' V (j3 e corr. m. 1), 
1*' m. 2 Vat. 19. forca (alt.)] om. m. 22. A] in ras. V. 

r] in ras. V, F sit&sla Vat.v. AF] seq. ras. 1 litt. V, 

AB Y. Dein add. kt di Vat.v. 23. ri AB] om. codd. 24. 
yavi&v t&v] om. Vat.v. i)it6] ycavi&v V7c6 Vat.v. ^tayo- 
^vmv] SucKSLiiivayv v. 26. Tj] r6 v. 26. yajviccg m. r&v] 
om. m. FAy AB] FAB m. 



i 




68 EUCLIDIS OPTICA. 

did Tov A si&eta ij ^AE. Xdjxo, Sri ^ i»tb FAB 
x^S i)Tcb EAB ilaeeav iavtv. %*w yaff &ab tov P 
inl tijv AE xa&ttog iv tm imaiSo) ij FZ, xal inE- 
gfti^^a) ^ BZ. xttl i] 
5 BZ eipa ixl ti)v ^E 
Xtt&srds itftiv. ixtl 
ovv 6q^ ^ ■bnb rZA^ 
■flimbArZ^QttiUa- 
Smv 6(f9^s- '^*' ^^ 

10 fisCtpva ycavCav ^ (iti- 
^mv jcXev^ic {moteivsi. 
Ij.si^t3v Spa ^ AV t^g 
AZ. &XX' ii ixb T&v 
AT, rs xal ii i)%b 

15 tStv BZ, ZA 6(f9ai 

elaiv raffrc slslv a[ FB, BZ aviSoi. xal ij vab t&v 
ZA, AB «pa r^s iijtb xStv FA, AB ioti fi£^£(av, bfioitog 
Si) Seij^&ijffitac xalxaOStv t&vyiavLmv tav 3t£QLSxo(tsviav 
vicb z(l>v Sitt rou A Siayofidvmv si&EiStv xal xotovaSn) 

20 «pog T^ AB Ev&eia ytoviav iXaxiCtri i} imh rav FA, AB. 

xttX <pkvsq6v, ort, iav Staj[&^ tig xal SXXt} sv&sta 

Sloi toC A &g ii A@ xoQ^aTS^ov ovOa t^g AF ^xs^ 

^ AZ, (isitav itstai 4^ i}%b BA& tijg imb BAZ. &x^Bi- 

«Tjg y&Q xdXiv xa&ixov iid tijv A& tfig FK ixi- 

25 ^%%slOa ^ BK x&&sxog iOtai biioitag Sxl tijv A@. 
xal insX (LEitfnv ii AA tijg AK (6p&i}v y&g imoxEivsi 
tifv i)7cb AKA), noXXm &^a ^ AZ t^s AK yi^itfsrv 



1. Sid] in ras. Y. toC] corr. es x6 xa. 2 V. rffl-ito] 

m. 2 Vat.; eniiiiov t, Tat. m. I. 2. iXdaaav] iXaxlavti Vm, 
lluMov T. 3. iJE] .iE T {in Vat. ^ ^ A diilicilliine digno- 
acantax). rro] im i:^ot.ti^ivta Vat.v. *. BZ] ZB Vatv. 



EUCLIDIS OPTICA. 69 

quod angalus gab angulo eah minor est. trahatur 
enim a puncto g super ae cathetus in subiacenti ebi- 
pedo, et coniungatur zh. zh ergo super ae cathetus 
est. quoniam ergo rectus gza, angulus ergo agz 
minor recto. maiori angulo maius latus autem sub- 5 
tenditur. maior ergo ag quam az, uerum anguli qui 
sub ag et gb et anguli qui sub hz et ^a recti sunt, 
et gh, hz inaequales. ergo qui sub za^ ah eo qui 
sub ga, ah maior est. similiter autem demonstrabitur 
et onmium angulorum contentorum sub ductis per a 10 
rectis et facientibus ad a& rectam* angulum minimus 
est gab. 

et manifestum, quoniam, si demonstretur et alia 
recta per a ut at remotior existens ab ag quam a0, 
maior erit hat quam ha0. tracta enim rursum catheto 15 
gk super at coniungatur hk cathetus similiter super at. 
et quoniam maior est alg quam afc; recto enim sub- 
tenditur ag^ multo ergo a0 quam ak maior est. et • 



3. coniugatur D. ergo] in ras. D. 5. autem] mg. 

m, 1 J). 7. anguli] anguli B. 16. coniungantur J). 17. 
alg^ 8cr. al. 18. ag] scr. akl. 



6. BZ] Z in ras. V. JE] AE Yat.v. 8. AFZ] corr. 

ex. A Y, AZ supra scr. B Vat.\ BAZ m. iXdeaovcc v, iXdt' 

roiv Vat. 11. {fTCotsLv^ v. 13. ^Z] BZ m. rj] ccl 

Vat.^m. M — 15. ZA] 'bnb AFB, BZA Vat.'m. 14. 

PB] r del. VVat. 15. BZ] Z del. Vat. Z^] Z del. V. 

16. slaLv] slai Vat.v. matB slaiv] om. VVat.*m, xa/ Vat.v. 

JBZ] BZ aga Vat.^m. t&v ZA, AB] ZAB m. 17. t&v 
TA^ AB] TAB m. iativ v. 20. AB(^r)] A m. y(oviag m. 

t&v rA,AB] FAB m. 21. q>avSQ6v] qp"^' Vat.\ (paaiv m. 
Svax^'^] SsLx^^ Vat.Vat.^mv (V?). 22. TtoQ^tSQOv Vat.v. 
23. ^isltov V. 24. tijs TK inl tijv A@ Vat.v. iyti- 

isvx^fj^cc V. 26. AA] corr. ex AJ V. 6q%t\ "^^ ^Qtss^»"^^» 



70 EUCLIDIS OPTICA. 

i6tiv. xal sttfLV d^d^al aC 'bTCo BZA^ BKA. iXd66(ov 
[ihv aQcc fj BZ TTig BK dcS^ rb t6a slvai tdc ts &7Co 
t&v BZ^ ZA %al ta aTcb t&v BK^ KA tfp Satb tfjs 
BA Tcal akXrlXoLg^ (iSL^iDv Se ndXiv f] imb BAK trjg 
5 ijtb BAZ, yta6a)v d^ t&v JtQbg tfj BA yLV0(iiva)v 
ya)vca)v iitb t&v diit tov A Svayoiiivoiv fieyL^trj i6tlv 
f} ijtb BAH ixpXrjd^ei^rjg r^g FA i^tl tb H^ ixal ocal 
7ta6&v iXdttG)v ij i)7tb BAF, t6ai S% yCvovtav at t6ov 
a%i%ov6at i(p* ixdtSQa trjg MA t7]g ti^v iXa%C6triv 

10 yG)vCav 7tSQtexov6rig iista tfjg BA. xsC^d^o) y&Q ty EM 
t6ri fi MN^ xal i7is^S'6x^a)6av at EM^ MN^ EF^ TiV, 
BE, BN^ AN. i7tsl ox>v t6ri i6tlv fj MN tfj ME^ 
xoLvij S^ fj MF, xal ymvCag t6ag 7tSQii%ov6vv^ t6rj aga 
xal ^ ET tfj FN. xocv^ Sh xal 7tQbg dQd^&g ^ FB, 

15 t6ri aga xal ^ EB tfj BN. aUct xal ^ EA tfj AN' 
xal xotvii ii AB. xal ya)vCa aQa ^ i)7tb EAB tfi imh 
NAB tarj i6tCv. 

"E6tai xvxXog b ABFA^ oi xivtQOv tb Z^ iv ^ 

svd^stav Vjx^^^^'^ ^^^ '^^^ ^? -S? ^9 -^ ts^vov6aL &XXifj' 

20 lag 7tQbg dgd^dg^ bii^a Sh i6ta) tb E^ d<p^ o5 ij i7tl tb 

xivtQov i7tL^svyvv^iv7] 7tQbg dQd^dg tfj F^^ 7tQbg Sl 



1. iatlv] iatL Yat.vm. BZA] ZBA Vat.v. §Xavtov v, 
iXdttGJV Vat. 3. BZ] ZB Vat.^m. r^s] t&v Vat. 4. 

iisliov V. Ss] corr. in &QCi V, &QOi Vat.Vat.*mv. 6. 6uc- 
yonsvcav] dttt- in ras. V. 7. iTtsl] xai insi Vat.v. 11. MN] 
corr. ex MF Vat. EF] om. m. 13. ytSQLSxovat v. 15. 
BN] BH m. 16. rj] tfjg Vat. 17. NAB] e corr. V. 

18. fty' V, ^a' m. 2 Vat. TistQov v. 19. rjx^oiGav] ^;^"^ 
Vat. &XXrjX(XLg m. 20. to iisvtQov] tov "iisvtQOv m. 21. rj] 
cojT. ex ^ Y. 



EUCLIDIS OPTICA. 71 

sunt recti hza et hTca [cum hza et hTca anguli 

triaugalis sunt recti^ tunc quadratum zh et ea ualent 
quadratum ha. similiter quadrata Jch, Jca ualent qua- 
dratum ha per elementa. et non igitur et inter se 
sunt aequalia^ cum ualeant idem. sed quadratum za 5 
maius est quadrato lca, quia za maior, sicut probatum 
est. ergo quadratum Jch est maius quadrato zh, ergo 
Jch maior zh']. minor ergo hz quam hh linea propter 
aequalia esse quae ab hz, za et ab hTc, Tca ei quae 
ab ha et ad inuicem. maior ergo rursum halc quam 10 
haz angulus [quoniam uero hag et hai anguli ualent 
duos rectos, similiter haz et had anguK ualent duos 
rectos. igitur ualent inter se. cum igitur hag sit 
minor haz, et hai erit maior had et sic de aliis]. 
omnium uero ad 6a factorum angulorum sub ductis 15 
per a maximus est hai educta ga super i, et quo- 
niam etiam omnium minor est hag. aequales uero 
fiunt aequaliter distantes ex utraque parte lineae ma 
minimum gah angulum continentis. iaceat enim rectae 
em aequalis mw, et coniungantur em, mn, eg, gn, he, 20 
ftw, an, ae. quoniam ergo aequalis est mn ei quae 
est me, communis uero mg^ et aequales angulos con- 
tinent; aequaKs ergo eg recta rectae gn. communis 
perpendicularis gh, aequalis ergo et e6 ei quae est hn, 
sed et ea ei quae est an. communis ergo ah. et 26 
angulus ergo aeh angulo nah est aequalis. 

Esto circulus aghd, cuius centrum z, in quo rectae 
trahantur per centrum ah, gd se ad inuicem perpen- 

1. Post anguli litt. quaedam duhiae D. 4. non] no D. 

18. inter se] his D, sed corr. 



72 



EUCLmiB OFriCA. 



vfjv AB XVX0V6KV ymviav mffuxhof xal ItSxta ^ EZ 
XTIS ix roC xivx(fOv fiei^cav. kiya, oxi ^vieoi aC Sid- 
(letffoi ccC AB, r.d ^av^Bovxtu^ xal ftfyitfti; (liv ij JT^, 
iXaxiex-ri Sl ^ AB, asl S\ ij Syyiov t^s ikaxisxTis 
5 iXdeeiav r^g aTiihxeQOv, S^^a S% pfvov SidftsxQoi iaai 




qsavTfSovxui lOov &nB%ov6ai itp' ixdxsQa xijg iXaxiSx'^^. 
i:id y&Q ii FJ ixaxiQa ratv AB, EZ iexi XQbg dQd^dg, 
xctl TiKVxa uQa xa 8ia x^g F^ iitCitESa ix^aXXdfiiva 
tip Sid z&v EZ, AB iexi icQog 6Q&dg' aexE xal xb 

10 ijxoxeCfitvov xov x-6xXov inCatSov, icp' oi iextv ^ FA. 
%^o) ovv &110 xov E eijiiiCov ial xb {moxsCfuvov ini- 
mSov xd%ixog. iiil tijv xoivijv iiQa roft^v ni-xxsi xStv 
ixiTtiSmv X7]v AB. jcmtixco ovv xal ieta ij EK, fcal 
dii]x6"M xfj diafiizQa xov xiixXov Ceti ^ AM xaX XE- 

16 Tfi^e&oi Sixa xaxk xb N er^^Etov, xal av^x%^c} anb 

tov N Ty AM HQog 6p9-^g ei&eia ij NS, xal lexa 

ri NS tfj EZ ietj. rb aQU nripi tijv AM yQafp6{icvov 

x(iflfitt xal iQx6(LBvov Si& tov S (let^^v ietiv ii^t- 

> xvxXiov, iatiSijm^ ij NS (i«t£a>v iexlv ixaxeQccs r&v 

20 AN, NM. iexes ro ASM, xal ixttevx^-ioeuv at SA, 



5. IXdveova t, comp. Vat. 
ii] poetea add. V, om. Vat.T 



djtrortpoij &}c6TtQ0v V Vat. v 
7. ynpj ow Vat.Vat."inT: 



EUCUDIS OPTICA. 73 

diculariter secantes. oculus uero sit e, a quo recta 
super centrum coniuncta ad rectos liueae gd, ad ab 
uero casu angulum contineat^ et sit ez ea quae a 
centro maior. dico^ quoniam inaequales diametri ab^gd 
apparebunt, et maxima quidem gd, minima uero ab, £ 
semperque propior minimae remotiore minor, duae 
tantum diametri aequales apparebunt aequaliter di- 
stantes ex utraque parte minimae. quoniam ergo gd 
utriusque a6, e0 est perpendicularis, et omnia ergo 
quae per gd ebipeda educta ei quod per eis, ab sunt K 
ad rectos. quare et^subiacentis circuli ebipedum, in 
quo est gd. trahatur ergo ab e puncto super sub- 
iacens ebipedum catbetus; super communem ergo 
sectionem ebipedorum scilicet ab cadet. cadat ergo 
et sit eJCy protrahaturque diametro circuli aequalis Im ll 
et diuidatur in duo aequa ad punctum w, et trahatur 
a puncto n rectae Im perpendicularis recta nx, sitque 
ea nx rectae esi aequalis. itaque circa Im descripta 
sectio ueniens per x maior est semicirculo^ quoniam 
recta nx maior est utraque nl, nm. esto Ixm, et 2( 
coniungantur Ix, xm. qui ergo ad x angulus con- 
tentus sub Ix, xm rectis aequalis est ei, qui est ad e 
punctum, contento sub e et 0, g, d. constituatur ad In 
rectam et ipsum n punctum angulus aequalis angulo, 



16. diametro] diametru, add. s, D. Mg. m. 1: ebipedum 
gd D. 17. a] ad D. 21. angulus] corr. ex angulos D. 



yccQ ovv, sed yaQ del., V. ianv v. 9. tc5] t&v m. 

iattv V. 14. dtifix^^] rJx^(o v. 19. iisltov v. iarlv] 

om. V. 



74 EUCLIDIS OPTICA. 

SM. fi &Qa TCQog xm S ycDvla fi xsQvexofisvrj ixb 
tcbv AtStj SM sid^sv&v t6ri i6xl tfj XQog rp E ^rjfisvp 
ty TtSQisxoiLivri imo tov E xal t&v F^ J. 6vvs6tcit(X) 
TCQog rg AN sid^sia xal tp N 6rjfisia} ty iico t&v 
6 HZy ZE t6ri ri imo t&v AN^ NO^ xal xsiad^io t^rj 
tfj EZ ij NO^ xal insis^x%^(x)6av at AO^ OM^ Tcal 
nsQLysyQaq)%^(o tcsqI to AOM tQtycovov tiirj^a to AOM, 
i^tau Sii %al fi XQog tp O ^rjfistc) ycovta t6rj ty JCQog 
• ta E tfi 'bTcb t&v HE®. iti 6vvs6t atco ^Qbg tfj AN 

10 si^d^sCcc xal Tco jCQbg aitfj 6rjfista} t& N tfj 4)7cb t&v 
AZE ycovta t6rj fj hnb t&v AN^ Nlt^ xal xsl6^g) 
tfj EZ t6ri fi NIl^ xal i7Cs^svx^G)6av a[ AU^ 11 M^ 
xal TCSQLysyQcifpd^a) jcsqI tb AIIM tQiycjvov tfifjiia 
xvkXov tb AIIM. i6taL Sij xal i^ ^Qbg rp 77 6rj(iSLCD 

16 yG)VLa t6rj tfj ijcb AEB ya)VL<f, insl ovv ^sl^cjv i6tlv 
fj XQbg rco tS! tfjg TCQbg ta O, (iAA' fj ^lv TCQbg t& tS 
^rj^SLO) t6rj tfj i)7cb FEA^ ij 6\ XQbg rc3 O trj i)7cb 
HES^ ^SL^a)v aQa q>avifj6staL rj Fz/ tfjg H®. tcccXlv 
iTCsl ij fihv JCQbg t& O 6rj^SL(p ycovta tfj inb HE® 

20 i6tLv t6rj^ rj Sh TCQbg t& 11 tfj {)7cb AEB^ ^st^a)v S' 
ij TCQbg rp O tfjg JCQbg rc5 77, ^sl^cov ccQa xal ij ijcb 
HE® tfjg i^cb AEB. ^sit^a^v ccQa cpav^^staL ij H® 
tfjg AB, 7Ca6&v ccQa t&v Sloc tov Z SLayoiisvcDV 
sifd^SL&v xal 7C0L0V6&V TCQbg trj EZ ya^vCag fisyC6trj 

26 fihv 6(p%^rj6staL ij Iz/, ikaxC6trj Sh ij AB^ S^dtL xal 
t&v JCQbg t(p E 6vvL6ta^sva)v ycovL&v iisyC6trj [lsv 
i6tLV ij vicb FEA^ iXaxC6trj Sl ij i)7cb AEB^ tfj Ss 



3. Tov] t6 V. 4:. AN^ AH m ras. V. iV] jtQhg 

ai^tjj m. x&v] om. m. 6. HZ] Z e corr. Vat. ifZ, Z£] 
EZH e corr. m. t&v AN, NO] ANO m. KsUd^co — 6. 
xa/ (pr.)] om. m. 7. AOM (j^r.)'] AEM y. XQiymvcp v. 



EUCUDIS OPTICA. 75 

qui continetur sub i0y ze, et contineatur ille angulus 
sub In, nOy et iaceat aequaKs ei quae est ee recta wo, 
et coniungantur ?o, om, et describatur circa trigonum 
lom sectio lom. erit autem et ad o punctum angulus 
aequalis angulo qui ad e sub iet amplius constituatur 
2Li. In rectam et ad ipsum punctum n angulus aze 
aequalis angulo Inp, iaceatque ei quae est ez aequa- 
lis npy et coniungantur Ip^ pm, et describatur circa 
Ipm trigonum sectio circuli Imp. esto ergo et qui 
ad p punctum angulus aequaUs ei qui sub aeh angulo. l 
quoniam ergo maior est qui ad x quam qui ad o, et 
qui ad x punctum aequalis angulo ged, qui uero ad o 
angulo iety maior ergo apparebit gd quam it rursum 
quoniam qui ad o punctum angulus angulo iet est 
aequalis, qui uero ad jp angulo aeh, maior uero qui l 
ad quam qui ad jp, maior ergo angulus iet quam aeh, 
maior ergo apparebit it quam ah. omnium ergo 
ductarum per z rectarum et facientium 2i,d. ez angulum 
maxima quidem uidebitur gdy minima uero ahy prop- 
terea quod et ad e constitutorum angulorum maximus 2( 

3. circa] contraD. 12. Mg. q2 D. angulo] angulusP D. 
13. it'] et D. 16. angulus] postea ins. m. 1 D. 19. gd] 
hd B. 



Tft^fta] V, m. 1 Vat., cxfi\La Vm, Vat. m. 2. 8. r©] x6 v. 

Gri\JLst Vat. 9. x&v] om. m. rj (alt.)] xov Vat.v. rj 
AN — 11. yaivla] in ras. V. 10. x^ (pr)] ^^^- ^^ ^^ ^^t. 

GTHLBiGi xai] corr. ex arnislov x6 Vat. x&v] om. m. 11. x&v] 
om. m. ' a'N, NU] ANII m. 12. rf] xfj v. 13. TtSQiYSYQcicpd-a)] 
om. m. x6 AIIM] xb A in ras. V. 14. 7iv%Xog Vat., corr. 
m. 2. ^Gxat] ^axa Vat.v. rd5] x6 y. 16. AEB] EB y. 

iaxl V. 16. S] Z vm. 17. 'Ol in ras. V, m. 18. if 
6 V. 21. xfjg] corr. ex rj m. 2 V, rj Vat.Wm. 22. xi^g' 
rj VVat.v, corr. m. 2 V. 24. r^] rfjg Vat.v. 26. r<3\ x6 y. 



76 EUCLIDIS OPnCA. 

d^sierjg a^rjg tfj HA trjg A T ocal iTCi^svxd^scdrjg trjg TZ 
Kal h^kri%^sl6rig iicl to H fi iTcb TEZ. tovto Sh 
Sflkov anb tSiv ^Qbg totg ^^ O^ U ycovi&v. %al y&Q 
5 tovt(x)v ikaxL6trj fihv ij U^ ixsl xal 'fj 'bicb UNA t6ri 
i6tl tri inb EZA ikaxC6tri ycDvCa^ [Lsyi6tri Sl ij S 
Sio: tb TCQbg dQd^i^g slvav tijv NtS ^syt^trjv yvvofisvrjv 
t&v Sv& tov N Scayo^ivcDV svd^SL&v iv tp AISM 
tfirliiatL xal tijv t^rjv avty tud^s^ivriv 'bjcsQTCiTCtSLV tb 

10 ASM tiiriiLa xal tb fihv S i^mtdto TCLTCtSLV tb Sh H 
i^(otdt(o ats (irjSsiiLag iXdttovog ytovCag ov6rig tr^g 'bnb 
HNA. tfig S\ 'b%b EZT t^rjg ov6rjg tfj 'bnb EZH^ 
iog TCQoSsSsLOCtaL^ Tcal ij i(ps^fig &Qa 'fi v%b EZ2J t6ri 
i6tl tfj 'bicb EZS^ tovti6tL tfj i)7cb ONM. &6ts sxa- 

15 tSQa t&v i)7cb TEU^ HES tfj TCQbg rp O l'6aL si6Cv. 
ij ccQa HS tfi T2J H^rj (pav^^staL, 

l6tc3 iXdttcjv ii aicb tov H^iiatog iicl tb xivtQOV 
ijCL^svyvviiivrj tfjg ix tofj xivtQov. akXd Sii jcsqI tdg 
SLafiitQOvg toivavtCov i^ yaQ TCQdtSQOv ^sC^cjv vvv 

20 iXd66cov (pavnl^staL^ ij Sh iXd66(ov ^sCtfov. i6tco x^vxXog 
6 ABFA^ xal SLtlx^co^av Svo SLdfistQOL aC AB^ FA 
ts^vov6aL aXXifiXag JCQbg dQd^dg^ itSQa Ss tvg tvxov6a 
Sli/jx^(o ii EZ^ 8/xfia Sl l6tG) tb 0, a(p' oi) ii iicl tb 
xivtQOv inLt,svx^si6a l6tc3 ij H® ikd66(ov ov6a sxa- 

26 tSQag t&v ix tov xivtQOv. xal xsC^d^co tf tov x^vxXov 



2. AT^ corr. &k AT Vat., AF m. iTtSLisvx^^nf^Sig v. 5. 
vn6] ccn6 v. 6. ^] Z m. 7. NlSl] NZ m. 8. AlSlM] 
AMZ Vat., sed corr. 9. a{)rfjg Vat.v. vTtSQTtiTttsi v, et 
Vat., sed corr. 10. TtLTttsi v. JT] in ras. V. 12. 

nNA] A in ras. V. £Z tfj 67ig v. EZH] ^ZH v. 14. 
ONM] OMN m. 16. HE@] in ras. V. iWt sldv] l^ari 



EUCLIDIS OPTICA. 77 

quideiD gedy minimus uero aeh^ angulo uero iet alius 
unus solus aequalis statuetur ablata aequali ei quae ia 
ab a^ et tz educta super s angulus tes. boc autem 
manifestum ab eis qui ad x^ Oy p angulis. etenim 
eorum minimus quidem p, quoniam et angulus pnl 5 
aequalis est angulo eea minimo angulo^ maximus 
uero X propter perpendicularem esse nx maximam 
factam ductarum per n rectarum in Ixm sectione, et 
aequalem ez eius positam et Ixm sectio supercadit, 
et X ualde extra cadito et p ualde extra uelut nuUo 10 
minori angulo existente angulo pnl. eo uero qui 
sub ezt aequali existente ei qui sub ezi. quare utri- 
que angulorum tes et iet ei qui ad o aequales sunt. 
itaque it ei qui est ts aequalis apparebit. 

esto minor ab oculo super centrum coniugata ea 15 
quae a centro. at uero circa diametrum e contrario. 
qui enim primum maior, nunc minor apparebit, minor 
uero maior est. esto circulus agbd, et protrahantur 
duae diametri ahj gd secantes se ad inuicem perpen- 
diculariter, altera uero diameter protrahatur e0f oculus 20 
uero sit t^ a quo super centrum sit it minor existens 
utraque earum quae e centro. iaceat enim circuli 



2. quae] qui D. 7.' propter — 8. sectione] mg. m. 1 D. 

9. aequalem] aequale D {quae seq., corrwpta). 10. ualde {dlt.)\ 
in ras. D. 12. ezf^ eis in ras. D. utrique] -ri- in ras. I). 

14. De schdlio hic inserto u. prolegom. it^ git D. 16. at] 
ad D. 17. nunc] nec D. 20. altera] alterai D. 



iativ Vat.^m. 17. (iS' V, fijJ' m. 2 Vat. iXdTtcov] in 

ras. V. 18. Post &XXd spat. uac. V. 20. ihdaaav (utr.Jl 

iXdttcov Vat.mv. 24. iXdttcav Vat., iXdttova v. 25. xatj 
om. m. 



78 EUCLIDIS OPTICA. 

ScaiistQG) t0ri 'fj KA xal rsrfii^dd^G) 8l%a yctiixk ro M, 
%al avJii^^fQ ^^^ '^^^ ^ 0rjiiSLOv ^Qbg dQd^&g fj MNj 
Tcal i6r(o t^rj ij MN rf} SH^ xal xsQiysyQag^d^cs xsqI 
rijv KA xal rb N ^rjfistov rfifj^a xiixkov rb NKA. 
6 i'0rL 8ii iXa66ov 'fj^txvxXiov^ iTtsiSifpcsQ ii MN ika00(Qv 
i6rl rrig ix rov xivrQOv. i6rav Sii TtQbg rc3 N ymvCa 
xsQisxofiivrj ijtb r&v KN^ AN t6rj rf} JtQbg r^ 6>, 
TtSQLSXoiiivri S\ iTtb r&v JT®, ®jd, Irv xsC^d^fo ry i^b 
r&v EH® t6ri rj imb rcbv KMS^ ^cci xsC^d^co rfj H@ 

10 l!6rj ij MS^ xal TtSQLysyQatpd^co xsqI rijv KA xal rb S 
6riiistov rb KSA rfifi^a. i6rvv &Qa ^Qbg rm tS 6rjfisCG) 
ycDvCa ij xsQisxofiivrj ijtb r&v KSA i'6rj rfj TtQbg rp ®^ 
xsQLSxo^ivrj Ss hnb rcbv ZSE. iri xsC^d^cD rfj ijtb 
rS)v AH^ H® t6rj ij ixb r&v KM^ MO^ xal xsC^d^o) 

16 ij MO rfj H® t6rj^ xal nsQLysyQdq^Q^co nsQl rijv KA 
xal rb O rfifjfia. i6raL Sij ij TtQbg rm O ycovCa jtSQV- 
sxofiivrj inb r&v KOA t6rj rfj TtQbg r^ S ycovCa TtSQv- 
sxo^ivrj ijtb r&v ASB. ijtsl oiv ^sC^g)v i^ ^Qbg rc3 O 
rfjg TtQbg r& S^ t6rj Sh ij ^lv JtQbg rS O rf XQbg 

20 ra ©5 TtsQLSxofiivrj Sh ijtb r&v A®B^ ij Sh XQbg r^ S 
rfj TtQbg r« 0, TtSQLSXofiivy Sh ijtb rcbv ESZ^ fisC^cjv 

1. $iccii,srQOv V, comp. Vat. M] Sfnia v. 3. tari] ^is v. 

GH] @N m. 4. KvtiXog v, comp. Vat. tb NKA] tb 

dl NKA Vat., x6 KA v, xh KNA m. 5. ^cm 6ri] Uxiv 6i v, 
%axC dl Vat. ^Xccaaov] ^Xccxxov Vat. MN] NM m. iXda- 
atov] iXdxxoyv v, IXaxxov Vat. 6. ^ivxQov] %v%Xov v, et Vat., 
corr. m. 2. ^axai] v, Vat. m. 1, for© Vm, Vat. m. 2. rco] 
corr. ex x6 V. 7. KN] JfiVf Vat.^m. AN] in ras. V, 

MNA V, et Vat., sed corr. 9. i^] om. v. x&v (alt.)] om. m. 

KMlSl] M in ras. V, KMZ Vat.^m. 11. aTHisiq)] om. m. 

13. Z@E] S@E V, et Vat., corr. m. 2. 14. x&v (utrum- 
que)] om. Vat.^m. AH, H@] AH@ Vat.^m. KM, MO] 
KMO Vat.^mv, et Vat., corr. m. 2. 16. O (pr.)] O arnislov 
Vat.^m. xiLfjiLa xvxXov Vat.^m. rco] x6 v. 13. 0] 

e corr. V. 19. xfjg — 0] bis m. 



EUCLIDIS OPTICA. 



79 



diametro aequalis lcl et diuidatur in duo aequa secun- 
dum sectionem. protrahatur a puncto medio perpen- 
dicularis mw, et sit aequalis mn recta rectae tiy et 
describatur circa lcl et n sectio circuli Jcnh est autem 
minor semicirculo, quoniam mn minor est ea quae e 





centro. erit autem ad n angulus contentus sub ifew, nl 
aequalis ei qui ad t contento sub gty td. amplius 
iaceat ei quae est sub eit aequalis sub kmXy et iaceat 
ei quae est it aequalis mx^ et describatur circa Jcl 
et X punctum klx sectio. est ergo ad x punctum lo 
angulus contentus sub Jclx aequalis ei qui ad t con- 
tento sub t^e, amplius iaceat ei qui sub ait aequalis 
qui sub JcmOy iaceatque mo ei quae est it aequalis, 
et describatur circa Jcl et o punctum sectio. erit 
autem qui ad o angulus contentus sub Jcol aequaKs 15 
ei qui ad t contento sub atb^ qui uero ad x ei qui 
ad ty contento uero sub et0j maior ergo apparebit ah 



9. quae] qui D. circa] in ras. B. 10. punctum (jw.)] 
puncto B. 11. Tclx] scr. kxl. 12. tze\ scr. etz. ei] mg. 
m. 1 D. 13. quaej qui D. IT. (kh\ mg. w.. 1. Ti. 



80 EUCLEDIS OPTICA. 

aQa <pavrl6€tai ii AB rrig' EZ, TCaXiv ijtel ^st^ov ij 
TCQog r« ® jteQLSXO^vrj ino t&v ES^ SZ rrjg XQog 
rc3 0, TtSQLexo^dvrjg Sh ijtb r&v FS^^ fisL^iov aQa 
6'q)%"if^6sraL ii EZ rf^g Tz/. 

6 Ag'. 

Tcbv icQiidrcov oC rQo^ol Ttore fihv xvxkoeLSetg (paC- 
vovraL^ 7tor% S% jtaQS^Jta^fisvoL, 

IdrcD rQoxbg 6 jiBF^^ xal Slt^x^^^^'^ SLa^srQOL 
at BA^ FA ri^Lvov^aL aXXifikag jtQbg dQd^&g xar& rb E 

10 6rj^SL0V^ xal xsl6%^(o Hfifia /xi) iv r& ijtLJtdSp rov xv- 
xXov. iocv ccQa ii a%b rov H^iiarog iitl rb xsvrQov 
iTtL^svyvv^ivrj ^Qbg dQd^ag ^ rp ijtLnsSc) tj i^6r] rfj ix 

^ rov xsvrQov^ a[ SLcifisrQOL 7ta6aL t6aL tpavifi^ovraL' 
&6rs 6 rQoxbg xvxkosiSiig <paLVsraL. iicv S% ii aitb rov 

15 bfiiiarog iTcl rb xsvrQOV iTtL^svyw^svri ^rjrs TtQbg dQd^&g 
^ r» inLniSco ^rjrs t6rj rfj ix rov xivrQOv^ at SlA- 
^srQOL avL60L ipavrl6ovraL j fiia iisv iisyL6rrj iiia Sl 
iXaxC6rri^ 7td6r] Sl akkrj ^sra^i) rrjg ^syL^rrjg xal rfjg 
iXaxL^rrjg SLriy^sini aXXri fita ^6vov dfpd^r^^sraL t6rj i%l 

20 roc ereQa iiiQrj SLrjyiiivrj' &6rs 6 rQoxbg 7taQs6Jta6fiivog 
cpaCvsraL. 

"E6rL rd^tog^ ov rov S^^arog ^svovrog^ rov Sh 6qc3- 

fiivov iisd^L^raiisvov^ t6ov dsl ro bQfb^svov (paCvsrai. 

26 s6rcD Sfiiia ro A^ dQ&^svov Sh fisysd^og ro BF^ dcp' 

ox) 7tQo6m7trsra)6av dxrtvsg al AB^ -^A ^^^ TtsQL- 

ysyQd^pd^a) xsqI ro ABF xvxkog b ABF. Xsyc^^ orL 



2. 'bnd] dh V7t6 m. E©, @Z] EGZ m. 0Z1 corr. ex 
&E Vat. 3. TCSQisxonsvri m. 5. Xg'] om. v, fis' V, fty' A, 



EUCLIDIS OPTICA. 



81 



quam ez, rursum quoniam maior qui ad t contentus 
sub et, tz eo qui ad t, contento uero sub tg^ tdj maior 
ergo uidebitur e0 quam gd. 

Curruum rotae aKquotiens circulares apparent, ali- 
quotiens parespamini. 5 

esto rota aghd, et protrahantur diametri ba, gd 
secantes se ad inuicem perpendiculariter ad e punctum, 

iaceatque oculus quidem in ebi- 
pedo circuli. si ergo recta ab 
oculo super centrum coniuncta 10 
non perpendicularis fuerit ebi- 
pedo nec ei quae e centro ae- 
qualis, diametri omnes inaequa- 
les apparebunt una quidem 
maxima, altera quidem minima, 16 
omnis autem alia inter maximam et minimam ducta 
alia una tantum uidebitur aequalis super alteras partes 
ductas. quare rota parespemenos. 

Est locus, in quo oculo manente eo, quod uidetur, 
transposito aequale semper, quod uidetur, apparet. 20 

esto oculus a, conspecta uero quantitas bg, a quo 
accidant radii »6, ag, et describatur circa ahg cir- 




2. tg'] corr. ex g D. 
cir- in ras. D, ut saepiics. 



12. quae] qui D. 22. circa] 



m. 2 Vat. 6. iiiv] jiri" Vat. 8. dLrjx&co Vat.v. 9. rsiivov' 
6iv A. 11. idv'] Bct' &v m. 12. Ttgdg] iirjts 7tQ6s Vat.A, 

ftr^ Ttgdg v. i) iari — 16. iTtiTtsdq}] om. codd. 16. ftifrg] 

in ras. V. farf] om. Vat.Av. ' rjl tf^g v. 17. &vL6ot] 

it&Gui codd. 19. ft/a] /t^v ft/a A; {lsv Vat., /t/a add. m. 2. 

20. 6] holI 6 Vat.A. 22. Xf'] om. v, /td' A; /ty' Vat. m. 2, 
corr. in ^8'-^ /tg' V. 

EuclideSf edd. Heiberg et Menge. 'VH. ^ 




82 EUCLIDIS OPTICA. 

i6tL tdytog^ oi (isvovtog iihv tov Hfifiatog^ tov Sh 6pa3- 

fidvov fieysd^ovg fisd^L^ta^ivov ^ l'6ov aal tb 6QG)fi6vov 

q>aCvetui, 

fied^i^td^d^co y&Q xal a6tc3 
5 tb Jr^ trj dh AB l'6rj l^tco 

fl A/1. inal ovv i'0r] i6tlv fi 

BA tfj AA^ fi 8% BF tfj Fz/, 

C^ri ccQa Kal ij BAT tfj AAR 

xal yaQ inl t^cov Jt£QLg)SQSLa)v 
10 sl0LV' &0t£ l'6aL sl6Cv, t6ov 

ccQa (pavi^66taL tb dQihiisvov. 

tb aiftb 8h ^vii^rl^etaLy %al sl tb S^^a inl tov 

xivtQov tov xtJxAov ^ivoL^ tb 8h dQmfisvov ijtl trjg 

jtSQLcpsQsCag iista^aCvoL, 

16 Xri\ 

"E6tL tLg tdjtog^ o5 tov Hfi^atog ^sd^L^ta^ivov^ tov 

Sh 6QC3fiivov iiivovtog^ asl t6ov tb 6q(6iisvov g^aCvstaL, 

i6ta) yd^Q 6qc)^svov ^sv tb BF^ H^^a Sh tb Z, ag?' 

ov jtQ06mjttita)6av ocxttvsg aC ZB^ ZF^ xal tcsql- 

20 ysyQdfpd^o) JtSQl tb BZF tQCyavov t^flfid tL xiixXov 
tb BZr^ xal fistaxsC^d^cj tb Z g/x/xc^ ixl tb A^ xal 
^sta7tLXtito6av al dxttvsg au AB^ AF, ovxovv i'6rj 

\ ^1 A ycovCa tfj Z' iv yaQ rw aitp t^7]^atC sC6lv, tcc 
S\ imb i!6G)v ycovL&v ^Qmiisva t6a cpaCvstaL, i'6ov aQa 

25 ro J5F Slu Ttavtbg cpavsttaL tov Sii^atog fisd^L^taiiivov 
ijtl tfig BAF TtSQLtpsQsCag, 



1. ^axiv Vat. 4. ya^] ycLQ xh BT Vat.^m; xh BF supra 

8cr. m. 2 V. 5. JF] J codd. $h AS] AA A. Uxai] 

itsxlv VVat.'m. 7. BA~\ A e corr. V. Ad] e corr. V. 

8. JAT] in ras. Y, AJF Vat.Av. 10. maxs tacci slalv] 



EUCLIDIS OPTICA. 83 

culus ahg, dico, quoniam est locus, jabi manente 
oculo conspecta magnitudine transposita aequale sem- 
per, quod uidetur, apparet. 

transponatur enim et sit d, et ei quae est ah 
aequalis esto ad. quoniam ergo aequalis est 6a ei 5 
quae est ad et hg ei quae est gd, aequaKs ergo et 
hag angulus ei quae est dag. etenim super aequales 
periferias sunt. aequale ergo apparebit, quod uidetur. 

idem autem continget, si oculus super centrum 
circuli maneat, quod autem uidetur, super circum- 10 
ferentiam uadat. 

Est locus, ubi oculo transposito, eo uero quod 
uidetur manente, semper aequale, quod uidetur, apparet. 
esto enim, quod uidetur, hg^ oculus autem z^ a quo 
accidant radii zh, zg, et describatur circa zhg tri- 16 

gonum sectio circuli hgz^ 
et transeat oculus z super d, 
et transcidant radii dhy dg, 
igitur aequalis d angulus 
angulo 8\ in eadem enim 20 
sectione sunt. quae autem sub aequalibus angulis uisa 
aequalia apparent. aequale igitur hg per totum ap- 
parebit oculo transposito super dhg periferiam. 

7. super] est super D. 20. eniiii] comp. mg. m. 1 B, 

sed del. 




om. m. 12. di] Srj Vat.Av. 15. Ztj'] om. v, fig' V, fis' 

Vat. m. 2. 16. ro-D (pr.)] rcJ vm. 20. BZF] ZBT Vat.v. 
Tfcl om. m, tov Vat. (corr. m. 2), v. 21. pbstccKslad^co] fifra- 
TiJ&fsd^a} m. 22. ^LSxaitimsxfQ v. al (pr.) — dT] xal ai 

dB, jr 6c%XLVsg m. 25. tpavi)tcci v. 26. BdF] T e corr., 
supra scr. Z V, BZJF Vat.^m, Z supra scr. m. 2 Vat. 



84 



EUCLIDIS OPTICA. 



'Eav fiiysd^dg xv TCQbg dQd^o^g ^ r& i)7toK€Lii6VG) im- 

TCsSp^ rsd"fj Sh ro '6^^a i%C ri ^rjnstov rov imjtsSov 

xal fisd^L^triraL rb 6q(oiisvov i%l xvxXov nsQLq>SQsCag 
6 TcivTQOv i%ovrog rb H^fiaj t6ov asl rb bQih^svov 

dcpd^ildsraL xaroL jcaQaXXrjXov d^i^LV r§ ^| ^QX^S fisra- 

Patvov. 

i6r(o 6qg)^sv6v rL ^iysd^og rb AB jCQbg dQd^&g 8i/ 

rp iiCLniSfp^ Hfi^a Ss i6r(o rb F. xal ijtsisrix^ci} i^ rs^ 
10 xal xivrQO} fihv rp JT, SLa6rri- 

fiarL Sh r^ FB xjixkog ys- 

yQccfpd^co 6 B^, Xiyco^ ort, ii:v 

i^l rrjg rov x^xXov TCSQLtpsQsCag 

^sd^L^rrjraL ro AB ^iysd^og^ 
16 &7cb rov r Sftfiarog H60V dfpd^Tl- 

6sraL rb AB, xal y&Q fi AB 

dQd^ i6rL xal noLSi sCQbg rijv 

BF yc3VLav 6Qd"i]Vj jca6aL Sh 

a[ aicb rov F xivrQov 7CQo6%L%rov6aL JCQbg rijv rov 
20 xiixkov TCSQLtpiQSLav sid^siaL t6ag ycjvCag jcolov6lv, t6ov 

aQa ro 6q(o^svov 6(pd"^6sraL ^iysd^og. 

i&v Sh ajcb rov F xivrQOv JCQbg ^Qd^o^g ava6ra%^ 

sid^sia^ i%l S\ raiirrjg rb ofifia rsd^fj^ xal fisraxLvfiraL 

rb 6Qm^svov fiiysd^og xara rrig rov xiixXov nsQL^psQsCag 

« 

26 %aQdXXriXov ^v ry si^^sCcc^ i(p* ^g ro oft/xo;, t6ov asl ro 
6q(oiisvov 6(pd"}j6sraL, 




1. X^'] om. V, fiTi' Y, fx-s' Vat. m. 2. 4. iisd^latatai 

Vat., corr. m. 2. 6. &sl Haov Vat.v. 10. tisvtqov v. 11. 
91 rai] corr. ex 81 x6 m. 2 V. FS] BF Vat.^m. 12. 6] 

^ Vat.v. 13. tfjg] om. v. rov] om. Vat., corr. in r^s v. 



EUCLIDIS OPTICA. 85 

quantitas aliqua perpendicularis fuerit sub- 
iacenti plano ebipedo, ponatur autem oculus super 
aliquod punctum ebipedi, transponaturque, quod uide- 
tur, super circuli periferiam centrum habentis oculum, 
semper aequaKs res conspecta uidetur secundum par- t 
allelam positionem ei quae e principio transiens. 

esto, quae uidetur, aliqua magnitudo ab perpendi- 
cularis existens ebipedo, oculus uero sit g^ et con- 
iungatur gh, et centro quidem g spatio gh circulus 
describatur hd, dico, quoniam, si super circuli peri- lO 
feriam transponatur ah magnitudo, ab oculo g aequalis 
uidebitur ah. etenim ah recta est et facit ad hg 
angulum rectum, omnesque a centro accidentes ad 
circuli periferiam rectae aequales angulos faciunt. 
aequalis ergo conspecta uidebitur magnitudo. 16 

si uero a centro g perpendiculariter consurgat 
recta, et super eam oculus ponatur, et transponatur 
conspecta magnitudo secundum circuK periferiam par- 
allelos existens rectae, super quam est oculus, aequalis 
semper res conspecta uidetur. 20 



5. aequalisl corr. ex aequales m. 1 D. 8. coniungan- 

tur D. 10. describitur D. 11. a&] u ah D. 20. con- 
specta] completa D. 



14. pbs&laTcitaL m, et Vat., sed corr. 18. dQd^ijv] om. Wat.* m. 
19. rov (pr.)] om. v, m. 2 Vat. %svtqov] hvtiXov Vat. 20. 

7toi.ov6i.v] jrotovcri"' Vat., noiovaai v. 21. xh iiiysd^og v. 22. 
TiivtQOv] corr. ex ^vkXov Vat. 23. \LSxccv,ivT)tai\ pbstaTiivsttccL 
Vm, et Vat.^, sed corr. 26. 7taqdU,r}siv (i-v^^Ta»^. ^. 



86 EUCLIDIS OPTICA. 

lidvG) invTtiSG}^ iisd^ierrjtai dl iitl tcvkXov TtBQKpBQsCag 
t6ov hv ifj ix tov xivtQov, sror^ iihv t6ov eavtip^ Jtoth 
5 dh avL6ov dq^d^T^estai xatic itaQdXXrilov ^i6iv tfj i^ 
&QXf}g {lata^alvov, 

i6t(o xiixkog 6 ^^, xal slXi/iq^^c^ iTtl tfjg tceql- 
(psQsCag aiftov erjiistov tb ^, xal i(ps6tdt(o iiii TtQog 
dQd^dig tp x^ixXco ax^d^eta fj dZ i!6rj oi)6a tfj ix tov 

10 xivtQOv^ o^iia Sh i0t(o ro E, liycj^ ort fj ^Z, iav 
ijtl tfjg tov xiixXov itSQKpsQsCag ^sd^C^trjtai^ Jtoth t^rj 
(pavifi6BtaL^ Ttota {leCtjcov^ itotl ild66c3v, fjx^^c^ Sij St,a 
tov E^ 8 i6ti xivtQOv^ trj AZ TtaQdkkrjlog fj FE^ xal 
i6t(o t6rj tfj ^Z ij EF. xal %'9'G} aTtb tov F 6r}^aCov 

15 iTtl tb i)7toxaC^avov iitCitaSov xdd^atog fj FH xal 6v^- 
Pakkito t& i7tL7tiS(p xarei ro H 6i^^atov, xa\ iitt,- 
^avx^at6a ij EH ix^a^Xifj^^^^o Tcal 6viL^aXXit(o tfj TtaQi- 

(paQaCcc xatic tb A 6rj^atov^ xal i^x^^ ^^& '^^^ ^ '^ll 
FE TtaQdllrilog ij AB^ xal i6t(o ij AB tfj JZ t^rj. 
20 Xiycj^ Srt fj AB Jta6a)v tG)V iitl tfjg tov xvxXov itaQL- 
(paQaCag fiad^c^ta^ivcjv ai^d^ai&v ikd66(ov (pavrj^atac, 
i7tata}ix^c36av yccQ aid^atai at EJ,, TZ, T^, EB^ ZE, 
ijtal ovv ij FE tfj AB TtaQdllrjUg i6tv xal t6rj, xal 
ij EA aQa tfj FB l'6rj ta xal jtaQakkrjlog i6tiv, TtaQ- 



1. ft'] om. V. fi^' V, ftf m. 2 Vat. 3. {LB^^iGtocxai Vat., 
corr. m. 2. 8b'] 8\ rf^s Vat. 7. AJ] inter A et J ras. 1 
litt. m. 11. Ttoth iiiv m. tari] ^^^'^ ^- ^^- ''^^^ ^* 

bis m. 13. xfiW^o) v. 14. EF] TE m. 19. ^Z] JS 

Vat. 21. iXdttoDv Ysit, Uocttov v. 22. EB] supra scr. V 
(Ez:/ — ZE etiam in mg. m. 1 V, TZ supra scr.). 24. rj 
-TjS apo( Yat.Ay. ictiv] icti Vat.Avm. 



EUCLIDIS OPTICA. 



87 



Si, quod uidetur, subiacenti ebipedo perpendiculare 
non fuerit; transponatur uero super circuli pariferiam 
aequale existens ei quae e centro, aliquotiens quidem 
aequale ei, aUquotiens uero inaequale uidebitur secun- 
dum parallelam positionem ei quae e principio transiens. s 

esto circulus ady et sumatur in periferia eius 
punctus dj et inde surgat non perpendicularis circulo 

recta dz aequalis 
existens ei quae e 
centro, oculus uero ic 
sit e, dico, quoniam 
dZj si in circuli pari- 
feria transponatur, 
aKquotiens quidem 
aequalis apparebit, it 
aliquotiens maior, 
aliquotiens minor. 
trahatur autem per e, 
quod est centrum, 
rectae dz parallela 2C 
g e , trahaturque a 
puncto g subiacens ebipedum cathetus gl et con- 
cidat ebipedo ad i punctum et coniugata ei edu- 
catur et coniungatur ad periferiam ad punctum a, et 
trahatur per punctum a rectae ge parallela ahy sitque 2fi 
recta ah rectae dz aequalis. dico, quoniam ah om- 
nium super circuli periferiam transpositarum rectarum 
minima apparebit. coniungantur enim ed, gz, ghy eh, 0e. 
quoniam ergo recta ge rectae ah parallelos existens 
est et aequalis, et recta ergo ea rectae gh aequalis 30 

22. gl] scr. gi. 25. paralellam D. 29. recta] rectam D. 




88 EUCLIDIS OPTICA. 

akkrjkdyQaiiiiov ccQa i6tl tb AEFB. 8icc tA avtct di) 
jtaQaXkrjkdyQaiiiiLdv i6ti xal tb EAZF, XbCtcbv 8h 

-\ Ssli^ai^ Stt sXa66ov (paCvBtai tb aitb zal iist^ov. fpa- 
VBQbv Sifi^ ott ild66ov B6tl ycjvCa 'fj ijtb FEA rijff 
5 'bicb FEA^ ijCBl dsdBiKtac^ oti Jta6&v tcbv 8idc tov 
xBvtQov diayo^BVCjv Bvd^Bc&v xal 7toLOv6&v ycjvCav 
ila%C6tYi i6tlv 'fj {jTtb FEA, iXa66c3v aQa i6tl xal 
. tflg {)7tb FEA. xaC i6ti tfjg iihv 'b%b FEA 'fi^C^Bta 
i^ 'bjtb BEA' jtaQalXt^loyQa^^ov y&Q l66jtXBVQOv tb 

10 BE' tfig 81 'bitb FEJ 'fj {)7tb ZEA' jtaQaXlifjldyQaii- 
liov y&Q C66^Xbvqov xal tb ZE, xal ij '^^^ BEA 
&Qa ikdttcjv i6tl tfig 'bjtb ZEA, &6tB xal tb AB 
^iysd^og tov AZ \iByB^ovg ikattov 6q>d^6Btai, 

xal (pavBQbv ix tov jtQodBdBiyiiBvov Ai^/xftarog, ort 

16 ikd%i6tov ^lv 6(pd^6Btai TtQbg tp A^ iiiyi^tov Sl jtQbg 
tp xatcc did^BtQov tp A 6riiiBC(p^ l6ov 8\ tb t6ov 
&7CB%ov i(p' BxdtBQa toi) A ^tj^bCov, 

\ka , 

'Eav 8b tb 6q6\ibvov JtQbg ^Qd^ag ?i ta {>7toxBv\iBVG> 

20 im7tB8c)^ ^iBd^C^tritai d^ tb '6\i\ia iTtl xvxXov jtBQi- 

(pBQsCag xBvtQov S%ovtog tb 6riiiBiov^ xad^' b 6v^^dkkBL 

tb iiiyBd^og rp i7ti7tB8(p^ t6ov aBl tb 6qg)\ibvov (pavr^'- 

6Btai, 

B6tco 6q6\ibvov ^iiyBd^og tb AB 7tQbg ^Qd^dg t(p 



1. icri] om. m. AETB] AEBT Ym. 2. TtaQaXXriXo' 
yQcciiiicc A, comp. Vat. tb EJZF] mg. m. 2 V. 3. oti,] 

mg. m. 2 V. ^Xocttov Vat., comp. v. 4. ^Xaaaov v, comp. 
Vat. 5. iTtsl] seq. ras. 2 litt. V, i^Jtsl ovv Vat.Av. 6. 

ycDviav] dgd^iiv ycoviav Vm, dqQ^riv add. m. 2 Vat. 7. FEA] 
TEA yoavia m. iXdttoDv Vat., comp. v. aQu iati] iatlv 

&QOC Vat.Av. 8. FEA] Am ras. V, TEz/ A; FEA v, et 



\ 



EUCLIDIS OPTICA. 89 

et parallelos est. parallelogrammuin est ergo aegh. 
propter eadem uero et parallelogrammum existit ed^g. 
restat autem demonstrare, quoniam minus apparet idem 
et maius. manifestum est autem, quod minor est 
angulus gea quam ged, quoniam ergo demonstratum 5 
est; quod onmium per centrum ductarum rectarum et 
facientium angulum minimum est quae sub gea, minor 
ergo quam ged. et est angulus quidem gei medietas 
angulus hea] parallelogrammum aequilaterum est enim. 
et 0ed medietas anguli ged'^ parallelogrammum enim 10 
aequilatferum est. et qui sub hea ergo minor est eo 
qui sub 0ed. quare et ah magnitudo magnitudine dis 
minor uidebitur. 

et manifestum est ex praeostensa ratione, quoniam 
minimum quidem uidebitur ad a punctum, maximum 15 
uero ad illud, quod secundum diametrum distat ab a 
puncto, aequale uero per aequale distans in utraque 
ab a puncto. 

Si; quod uidetur, perpendiculare fuerit subiacenti 
plano, transponatur uero oculus super circuli peri- 20 
feriam centrum habentem punctum, secundum quod 
coniungitur magnitudo ebipedo, aequale semper, quod 
uidebitur, apparebit. 

esto conspecta magnitudo ah perpendicularis sub- 

2. paralellogramum D, ut Im. 9, 10. 8. ergo] in ras. D. 
angulus] scr. anguli. 10. et — 11. est (pr.)] mg. m. 1 J). 
11. 60] his, sed corr., D. 17. utroque D. 

Vat., corr. m. 2. 9. BEA] BEF Vm. 11. .ya^] om. 

Vat.Av. 12. lcXattov v, sed corr. 17. i(p'] ictp' v. 18. /ia'] 

om. V, v' V, li/ri' m. 2 Vat. 20. iitl'] inl tov A, tov supra 
scr. Vat. 21. %%ovta v. o] in ras. V. 



90 



EUCLIDIS OPTICA. 




H60V &Qa ro 6^(6- 



'b7tox£L(iivp iytLTtdSa)^ oftfta Sh i6t(o ro JT. xa\ xevrQO) 
^lv rdi Bj Svaerij^arv Sh r^ BF xvzkog ysyQaq^d^Gi 
6 r^. kiyG)^ orv^ i&v iLe^C- 
6rrirai rb F ijcl rflg rot) xvxXov 
6 7ceQi(peQeCag^ l'6ov ael ro AB 
(pavT^^erai. rovro Si (pavegdv 
i6riv. 7Ca6av yccQ aC aTtb roi> F 
6rjiiLeCov JtQbg rb AB jtQ06- 
7tC7trov6ai axrtveg TtQbg l'6ag yc^- 
10 vCag 7tQo6jtCjtrov6LV^ ijteLSiJTteQ ^ ^ 
'fj TtQbg rco B ycovCa 6Qd"tl i6rvv, 
lievov 6(p%"/i6erav. 

Tov 6p(Oft6Vov iiivovrog^ rov Sh o^fiarog fied^v^ra- 
16 iiivov xar' eifd^etav yQa^^^^v itXayCav TtQog ro 6pc5- 
lievov ^iyed^og oi6av sror^ iiiv l'6ov^ jtorh Se avv6ov 
ro 6Q(x)(ievov (paCverav, 

i6r(o 6Q(X)(ievov (lev ro AB^ [o(i(ia Sh rb E^ eid^eta 
Sh TtXayCa i\ JTz/, xai TCQO^ex^e^hlfi^aQ rr^ BA in^ 
20 ei)^eCag 'fj FA xal 6v(iPakkir(o rfj ^F xara rb JT, 
xal (led^v^rd^d^co ijt^ airfjg ro b^i^ia, kiycj^ ort ^ror^ 
lihv t6ov^ 7tor\ Se avv6ov (paCverav ro AB. eCkij^pd^co 
yciQ rcbv BF^ FA (li^rj &vAXoyov fj FE^ xal i6rc3 
b(iiia ro E xai iieraxexvvT^^d^co xal S6rc3 ijtl rfjg airfig 
25 eid^evag xara ro A. kiycj^ brv ro vjtb r&v E^ A 6q6- 
(levov avv6ov (paCverav, i7te^evx^(X)6av evd^etav av AE^ 

1. 716VTQOV, corr. m. 2, Vat.A. 2. B] A Vat.Av. y«- 
ypa^-O-o)] 6 y£ y^agj-O-ai Vat., sed corr.; 6 yhyQat^^ta v. 4. xov\ 
om. V. 7. iffTt Vat.mv. 10. TtQoanLTtrovaoa v. 11. BJ 

corr. ex F Vat. 13. ^/3'] om. v, va' V, (id'' m. 2 Vat. 16. 
T(J] ro5 V. 17. (palvstai tb dgaiiisvov m. 18. ^fV] om. v. 



EUCLIDIS OPnCA. 



91 



iacenti plano, oculus uero sit g, et centro quidem 6, 
spatio uero hg circulus describatur gd, dico, quoniam, 
si transponatur g super circuli periferiam, aequalis 
semper ab apparebit. hoc autem manifestum est. 
omnes enim a puncto g ad ab accidentes radii ad 5 
aequales angulos accidunt^ quoniam qui ad h angulus 
rectus est. aequalis ergo res conspecta uidebitur. 

Re conspecta manente, oculo uero transposito se- 
cundum rectam lineam obliquam ad conspectam quanti- 
tatem existentem aliquotiens quidem aequalis, aliquo- 10 
tiens uero inaequaKs res conspecta apparebit. 

esto, quod uidetur quidem, ahy oculus autem sit e, 
recta uero obliqua gd, et adiciatur ei quae est &a in 

directo ag et con- 
iungatur rectae dg 16 
ad gy et transpona- 
tur oculus. dico, 
quoniam aliquo- 
tiens quidem aequa- 
lis, aliquotiens uero 20 
iuaequalis apparet 
ah, sumatur enim rectarum hg^gd media proportionalis 
ge, et sit oculus e et transmqueatur et sit in eadem 
recta d, dico, quod sub e, d uisum inaequale apparet. 




6. qui] g* D. &] ras. 1 litt. D. 13. oblique D. ba\ 
supra scr. m. 1 D. 



^ 81 hta Vat.v. 20. FAjABy, AT Vat. 21. a-Srflffl 
comp. Vat., oc'bx& v. 23. Post yag ras. 2 uel 3 litt. V. JBrj 
JBiVv. ' 



92 EUCLIDIS OPTICA. 

EB^ A^^ B^^ xal TtSQLysyQoig^d^io xsqI tb AEB tQC- 

y(ovov tiifjiiLa tb 'AEB^ xal X6i6d'(x) tfj i)%b t&v Fjd^ jdB 

yG)vCa t6ri ycovCa fj 'bTtb tmv FA^ AZ^ xal stcs^svx^co 

'll BZ. iv xiixXG) ccQa i6tl t& B^ A^ Z^ ^ 6rj^sta. 

5 ijtsl ohv iisCtcov ycovCa ii iTcb AEB tfig i)7tb AZB^ 

fj Sl imb AZB tfj i)icb t&v A^^ ^B t6ri i6tCv^ iicsi- 

"' di^jtSQ iv tp ai)t^ t^i^fiatC i6tLV^ ;cal ii i)7tb AEB 

&Qa tfjg i)7tb AAB ^nsCitov i6tCv. aiX* i)7tb iihv tfjg 

, i)7tb A^B tb AB ^XsTtstai tov o^iiatog iitl tov A 

10 ovtog^ i>7tb dl tfjg i)7tb AEB tb ax>tb tb AB ^XsTtstai 

tov o^iiatog iitl tov E ovtog, &vl6ov ccQa tb 6q(o- 

^svov (paCvstat iitl tfjg E^ si)^sCag rov o^iiatog 

. iisd^L^ta^svov, (p(3CVSQbv di^ Zti xal iitl tf]g E F iisd^c^ta- 

^svov tov b^^atog &vl6ov tb 6q(o^svov (paCvstat xal 

15 iiiyL6tov (ilv xatcc tijv TtQbg tm E d^i^LV^ ^st^ov Sh 

&sl xat& f^v iyyvtSQOv ai>tov i(p' b%otSQa6ofjv tcbv 

EA^ EF svd^SLcbv^ i'6ov Sh xatdc t& Z xal A xal tdc 

b^oC(og ai)totg Xa^^avd^sva Sict ro iv rc5 avrco rftiy- 

^atL slvaL tag ycovCag. 

20 "Aklc^g. 

"E6ta} ydcQ 6q6^svov tb K^^ svd^sta Sh ri BF 6vii- 
jtC7ttov6a tfi K^ 7tQ066xPakko^ivrj. slk^^^pd^a) tfjg Fjd 
xal tfjg FK iiLb6rj avaXoyov ij FZ^ xal ijts^svx^c^ ij 
ZK xal ij ZA^ TtSQl Ss f^v Kd t^fj^a ysyQcc^pd^cj^ b 



2. AEB] corr. ex AEH Vat. r&v] om. m. FJ, JB] 
rJB m. 3. T&v] om. m. FA, ATATAZ m. 4k. ij BZ] 
in ras., seq. ras. 2 litt., V, post ras. 3 litt. v. 5. fisttov v. 

6. r&v] om. m. Ad, JB] AJB Vat.^m. 7. iatLv] slat m, 
iiftL Vat. 7. 8. nstiov v. iarl v. 9. rb AB] om. codd. 

pXinstai rb AB m. rov (alt.)] rd m, 10. vTtb Sl tf}s] 

bis V. tb ccM tb AB pXsTtstccC] om. v. AB] A v, et Vat., 



EUCLIDIS OPTICA. 93 

coniungatur ae,e}), adjhd, et describatur circa aeh tri- 
gonum sectio aeh, iaceatque ei qui sub gd, hd angulo 
aequalis angulus qui sub ga, az, et coniungatur hz, in 
circulo ergo sunt h, a, 0, d puncta. quoniam ergo maior 
angulus aeh angulo aish, angulus uero azh ei qui sub 6 
ad, db aequaUs, quoniam in eadem sectione sunt, et 
angulus ergo aeh angulo adh maior est. sed sub 
angulo quidem adh uidetur ah oculo super d ente, 
sub angulo uero aeh idem ah uidetur oculo super e 
existente. inaequale ergo uisum apparet super ed 10 
rectam oculo transposito. manifestum uero, quoniam 
et semper super eg transposito oculo inaequale, quod 
uidetur, apparet, et maximum quidem.quae ad t positio- 
nem, maius uero ad ei propinquiorem in utralibet ergo 
ed, eg rectarum, aequale autem quae ad ea quae ad 16 
et quae ad d et ea quae simiKter ei sumpta propter 
in eadem sectione esse angulos. 

Esto enim, quod uidetur, Jcd, recta uero hg con- 
cidens ei quae est Jcd eductae. et sumatur rectae gd 
et rectae gJc media proportionalis g0, et coniungantur 20 
0Jc et 0d, et circa uero Jcd portio describatur circuli, 



3. coniungantur D. 12. inaequale] corr. mg. m. 1 ex 

aequale D. 13. quidam D. 16. et (pr.)] ea p)st ras, 1 
litt, D. 21. portio] corr. ex proportio D. 



corr. m. 2. 11. rov (alt.)] t6 m. &viaaov v. 12. r^g] 

Tov^ eras., v. 14. aviaaov v. 15. rw] corr. ex x6 v. 

16. oiiorBQaaovv'] -aa- in ras. V. 17. J] xk A Vat.v. 18. 
cL^oli\ ccbtov V. 20. &XX(os] om. Vat^m. 21. v§' V, v' 

m. 2 Vat. 23. FK fisari ^^"] i^ ^^^s. v. iTCsiB^bx^faauv 

Vat. (corr. m. 2), v. 24. xiqv'] r^s v. 




94 EUCLIDIS OPTICA. 

dixstai tiiv {fxb r&v KZ^d, itpailfBtai Sii tijg BF 
sifd^stag^ iTCet.SifiitBQ iog i} KF ytQog t^v TZ^ ovtmg i\ 

rz xQog tiiv r^. 

KBLed^o) di) to 3/xfea 
5 ijtl tov B 6rj^£ov^ 

Xal TtQOOBX^B^klfl- 

^d^fo^av av ^B^ 
BK. iTtBtB^^xd^co dh 
ij U^, ovxovv t6ri ^ ^ 

10 ii O ycovCa tfj Z! ycDvia' iv yd^Q t^ ax>t& t^nriiiatC 
bC6lv. xaC i6tLv ij 2J tfig B ycovCag ^bC^giv' xal ij 
' O ccQa ycovCa tfig B iibC^cov ietCv. tov ccQa S^^atog 
i%l rov Z '6vtog ^t^ov (paCvBtai to K^ ^jtBQ iitl roi) B. 

liy. 

15 To d' a^dro 6v^P'^6Btai^ xctv ytaQdklrjXog y ii Bid^Bta 
yQaii^ij ta bQco^iivco ^Byid^BL 

i6tco 6qg)^bvov ^iyBd^og tb AB xal tBt^rj^d^co 8C%a 
xata tb E 6rj^Btov^ xal avT^x^^ ^^^ ^^^ E tfi AB 
^Qog dgd^itg ij EZ^ i(p^ ^g 8ft/xa XBC^d^co tb Z, xal ijtB- 

20 t^Bvx^^Gieav Bi^d^Btav at ZA^ ZB^ xal TtBQiyByQccq^d^co TtBQl 
tb AZB tQCycovov t^riiia tb AZB^ xal i^x^^ ^^^ 
tov Z tri AB TtaQaXkriXog ij ZA^ xal ^BtaxBCad^co tb 
Sfifia iitl tb ^5 xal 7tQ067tvjttitco6av axttvBg aC A^^ 
^B. Xiyco^ ort ccxb tcbv ^, Z avL6a (pavT^^Btac. iTtB- 

26 ^B^^x^c^ ij AH. iTtBl oiv t6ri ycovCa ij iitb AZB tfj 



1. dsxstaC] 6vvB%6toci codd. rtjv] om. codd. x(bv\ rov 
codd. 8ri] in ras. V. 2. KT'] F in ras. V. 3. TZ] 

in ras. V. F/f] in ras. V. 6. TtgoasTipspXi^ad^o} y, et Vat., 
corr. m. 2. 10. 27] corr. ex P m. 2 Vat, 11. slai vm, et 
Vat., corr. m. 2. JB] post ras. 1 litt. V. 12. iisltov v. 

Jffrl Vat.vm. 13. rov (alt.)] r6 v. 14. fiy'] om. v, vy' V, 



EUCLIDIS OFnCA. 



95 



quae continebitur sub Tczd. contingetur autem ab hg 
recta^ quoniam sicut Tzg ad gz, ita gz ad gd, iaceat 
uero oculus super h punctum, et adiciatur dh rectae dTc, 
coniungatur autem sd, igitur aequalis f angulus 
angulo s; in eadem enim portione sunt. et est 5 5 
angulus angulo 6 maior. et f ergo angulus angulo h 
maior est. oculo ergo super z existente maius ap- 
paret lcd quam super h, 

Idem autem contingit, et si parallelos fuerit recta 
linea ei quae uidetur magnitudini. 10 

esto quae uidetur magnitudo ah et diuidatur in 
duo aequaKa ad e punctum, et protrabatur ab e magni- 

tudini ahe perpendicularis 
eZy in qua oculus z iaceat, 
et coniungantur za, zh, et 15 
describatur circa azh tri- 
^ gonum portio azh, et tra- 
hatur per z magnitudini ah 
parallelos zdy et transeat 
oculus super d, et accidant 20 
radii ad, dh, dico, quoniam a punctis d, z inaequalia 
apparebunt. coniungatur ai, quoniam ergo aequalis 

Fig. falsam, quam e V dedi Studien p. 121, corr. Weissen- 
bom Philol. XLV p. 57. 




4. coniungantur D. 9. fuerit] facit D. 12. magni- 

tudine D. 14. ez\ zez D. 22, coniungantur D. 

va m. 2 Vat. 16. {LsyiO^Ti ^- 20. xal nBQiy. — 21. AZB (alt.)] 
mg. m. 1 m. 21. -4ZB(pr.)] -4Z V; r/i^/ia] rft^fta nvnXov 
Vat.^m, in mg. add. -kMov m. 2 V. 22. fisra-] in ras. v. 
23. 6Va V. rd] rov Vat.v. 25. AH\ia ras. V, 



96 



EUCLIDIS OPTICA. 



{)7tb AHB^ aXl' ii {}7cb AHB trlg 'b%b AAB ii€Lt(ov 
B6tCv^ Kul ii ijcb AZB &j^ trlg 'bitb AAB ^ec^cov 
i6tCv. xal iitb [ihv tfjg 'bicb AZB tb AB ^kBTCstai 
tov Sftftatrog iTcl tov Z '6vtog^ b^oCcog dh zal vTcb tfjg 
5 ijtb AAB iitl tov A Svtog. &vl6ov &Qa ro bQ(o^6vov 
(paCvstai djtb t&v A^ Z. 

xal iav tsd'^ l'6rj tfj AZ ij ZF, EXattov ^sv xal 
ScTtb tov r (paCvetac i^jtsQ djtb tov Z, &7tb dl t&v JT, A 
H60V. 

10 iid\ 

El6\ tdTtoc^ i(p^ ovg tov ^^^atog iietatL^siiivov ta 

t6a ^ayi%"ri xal xoLV&g d^jtoXa^dvta td^tovg tvvctg Ttot% 

^\v t6a^ 7tot\ d^ &vi6a (paCvat ai. 

l6t(o S^iia iihv tb ©5 ^ayi%"ri 8% td AB^ BF^ xal 
16 i^x^^ ^^^ ^^^ ^ ^Qbg dQ^ag ii BZ xal itQO^aTt^a- 

pirj^d^cj iitl tb A. (pavaQbv 

8ifi^ 5tL xad"^ bjtOLOVovv tfig 

ZA [liQog av tad^fj tb 3/xfto;, 

rct: AB^ BF l'6a (pavifi6atai. 
20 (lataxaC^d^c^ 8ii tb S/xfia xal 

l6t(o tb E. liy(o^ Stv ScTtb tov 

E &vv6a (paCvatav, jtQ067tv7tti' 

t(o6av axtvvag at AE^ EB^ 

EF^ xal TtaQvyayQci^pd^co TtBQl tb AFE tQCyavov 6 
26 AEAF xvxkog^ xal TtQO^ax^a^^^^^cj tfj EB fj BH, 

iital ovv t6ri ij AA itaQV^piQava tfj AF TtaQv^pagaCcc^ 

[laC^cov 81 ij AAH 7CaQV(piQava tfjg HF 7taQV(paQaCag^ 




1. fislSov V. 2. iaTL Vat.vm. iist^ov v. 3. i6ti 

Vat.vm. -^Tro (alt.)] om. v. 4. xaH xal rj v. 5. 'bnd] 

del. m. 2 Vat., om. Vm. 6. &^6] vno codd. 7. ri\ tfj v. 

iXatttov m. 8. T (pr.)] N v. 10. ^d'] om. v, v8' V, v^' 



EUCLIDIS OPTICA. 97 

angulus qui sub azb angulo aih, sed aih angulus 
angulo adh maior, et angulus azh ergo angulo adh 
maior est. et sub angulo a^h magnitudo ah uidetur 
oculo super existente, similiter autem et sub angulo 
adh super d existente. inaequale ergo, quod uidetur, 5 
apparet sub punctis d^ 0. 

et si ponatur aequalis ei quae est d0 ea quae 
est g0f minor utique ah sub g apparet quam ah sub 0, 
a punctis uero g, d aequalis. 

Sunt loci, in quibus oculo transposito aequales 10 
magnitudines et communiter occupantes locos quosdam 
aliquotiens quidem aequales^ aliquotiens inaequales 
apparent. 

esto oculus quidem d, magnitudines ah, hg, et 
protrahatur a puncto h perpendicularis d0 et iniciatur 15 
super 0. manifestum autem, quoniam secundum quam- 
cunque eius quod est d0 partem si ponatur oculus, 
ah, hg apparebimt aequalia. transponatur autem oculus 
et sit e. dicO; quoniam a&, hg inaequalia apparent. 
accidant radii ea, eh, eg, et describatur circa aeg tri- 20 
gonum aedg circulus, et adiciatur ei quae est eh 
recta hi, quoniam ergo aequalis ad periferia gd peri- 
feriae, maior uero ai periferia quam ig, maior ergo 

8. a& (iw.)] zal) JD. 12. quidam D. 18. a6] nah B. 
aequalia] mg. m. 1 JD. 

m. 2 Vat. 14. &\ in ras. m. 2 V, -4 v, et Vat., corr. m. 2. 

ra\ x6 codd. ^B, BT] in ras. V. 15. BZ] B e corr. V, 
z^Z V, et Vat., sed corr. m. 2. 16. iitl x6'\ corr. ex Scjib xo^ 
m. 2 V. J\ Z V, et Vat., corr. m. 2. 18. &v\ idv codd. 

19. Br\ e corr. m. 2 Vat., FJ Vmv. 26. JF] FJ vm. 

27. AJH] AJ m. 

^nciid«8, edd. Heiberg etMengo. "VH. '^ 



98 EUCLIDIS OPTICA. 

^si^cov &Qa (pavri6etaL ij AB trlg BF. zav ybSta^aCvri 
d^ iTtl tfjg EH^ avL6a 6iiOL(og q)av7]6£tai^ zal i%l tcbv 
tov Kvxkov ^EQ&v X(OQlg tfjg JtQbg dQd^d^g i&v ted^f;^ 
avL6a (paivEtav^ xal i&v ixtbg tov xvxXov ted^fi iiij 
6 i^' ex^d^ecag ov tfj ^Z, avi6a (paCvetai, 

"AkXcog, 

*'E6t(X) yccQ i'6rj ij BF tfi jn^, Tcal iceQl ^lv tijv BF 
fHLLK^^TiXiov yeyQdq^d^cj tb BZF^ jteQl dh tijv F^ iiettov 
'fjliLxvKXCov tb rZ^' xal (paveQ6v^ orfr te^et tb tcqo- 

10 SLQrjfievov ij^iKvxkLOv, Svvatbv de i6tiv iitl tfjg Fjd 
yQciipav r/x-^fta [let^ov ijiiLxvxkCov. iav yicQ iTtod^d^fied^a 
d^etciv tiva ycjvCav^ dvvatbv fj^tv i6tLv iTtl tfjg JTz/ 
yQdtl^av t^fjlia x^^xXov de^^^evov ycjvCav t^rjv tfj iito- 
xeLiievrj d^eCcc ycovCcc^ hg icxb rot) Xy' tov tQCtov tcbv 

15 imjteda)v^ xal e6tav ro 6vvL6td^evov iTt' aitfjg ^et^ov 
ij^ixvxXCov^ &g aitb rov Aa' tov tQCtov t&v imTtedcDV, 
xal iitet^eiixd^Gi^av at BZ^ ZF^ ZA, oixovv rj iv tp 
ijliLXvxXCco ycjvCa iieCtjDov i6tl tfjg iv rc5 ^eCtjovL t^Tj- 
liatL, td Sh {)7tb ^eC^ovog ycjvCag 6Q6^eva ^eC^ova 

20 (paCvetaL' ineClcov aQa rj BF tfjg Fz/ (paCvetaL, '^v S% 
xal i'6rj. e6tLv aQa tdjtog xoLvog^ iv S ro ofifia idv 
ted^fj^ avL6a (paCvetaL td t6a. t6a Si (pavrj6etaL^ iiteL- 
Sdv iitl tcbv f £| aQxfjg 6rjfieCG)v fj tav inl tcbv BF^ 
rd iieLtfivcov rj^LXvxlCcov. 

3. x^Q^s] X Vat., ^jrco^/coi; v. 6. aXXcog^ Vat.v, om. m, 

vs' V, vy' m. 2 Vat. 8. rjiicKVKXlov v. rrjv'] tfjg v. fiat- 
i(ov V. 9. raftfit] rs iLstSov m. 10. ds] rs m. ttJs] rov v 
et comp. supra scr. Vat. FJ] corr. ex z/F m. 1 Vat. 11. 
yQciifjcci, — 12. rj] bis m, corr. m. 2. 12. Svvarov — 16. 

iitiTCsdcov] male del. Weissenbom 1. c. p. 58. 14. Xy'] in 

ras. V, X in ras. m; z/JT v, et Vat., corr. m. 2. 16. Xa''\ Xy' 
in ras. V; Zy'^ X m ras., m; z/F v, Vat. m. 1, Xy Vat. m. 2. 



EUCLIDIS OPTICA. 99 

apparebit ah quam bg, et si transeat oculus super ei, 
inaequalia similiter apparebunt, et super circuli partes 
seorsum perpendicularis si ponatur, inaequalia apparent, 
et si extra circulum ponatur non in directo existens 
ei quae est dis, inaequalia apparent. 5 

Aliter. 

esto enim aequalis hg ei quae est gd, et circa 
quidem hg semicirculus describatur h^g, at uero 
circa gd maior semicirculo gisd, et manifestum qui- 

dem, quoniam maior 10 

praedicto semicir- 
culo. possibile super 
gd scribere portio- 
nem maiorem quidem 
semicirculo. si enim 15 
supponamus acutum 
aliquem angulum, possibile est nobis super gd scribere 
portionem circuli continentem angulum aequalem sub- 
iacenti acuto angulo, ut habetur in IIP elementorum. 
et coniungantur h0, zg, zd. igitur qui in semicirculo 20 
angulus maior quam alius in maiori portione. sub 
maiori autem angulo uisa maiora apparent. erat autem 
aequalis. est ergo locus communis, in quo oculus gi 
ponatur, inaequalia apparent aequalia, quoniam quidem, 
si super ea quae a principio puncta fuerit, earum quae 25 
sunt hg, gd maior semicirculus. 

6. quae] qui D. 17. possibile] possi- seq. ras. 1 litt. D. 

17. Zr, Zz/] r, Z in ras. V. 18. fte/fcov] iist^ov v. 2Q. 

ILSt^ov y. (isl^av — (palvsraL] om. m. 21. notvoDg T. iH|| 
arnjbstov Yat.^ (^ m. ^^^^1 





100 EUCLIDIS OPTICA. 

"E6tL tvQ t6icoQ xoLvdg^ &g)' oi t& &vv6a iiL£ysd'rj l'6cc 
(paCvstav, 

i6tG) y&Q iiec^cov ij BF trjg JT^, xal jtSQl ^hv ti^v 
6 Br ^st^ov ijiiLLXvxkCov tiififia ysyQdfpd^co^ jcsqI di f^v 
r^ oiiOLOv t& TtsQl tiiv BF^ tovti6ti dsx6(isvov yco- 
vCav t6riv ty iv trp 

BZr. tSllL0V6LV CCQa 

alXYiXa t& t^iLYiiuxxa, 

10 ts^vdtc()6av xatct tb 
Z, xal S7Cslsv%^G}6av 
aC ZB^ Zr^ ZA. 
Ofbxovv STCsl t6at 
sl6lv at iv totg byLoCovg t^^^fia^c ycovCav akkifikaLg^ t6aL 

15 sl6l xal at iv tolg BZF^ FZJ tfiT^^a^L ycovCaL aXXri- 
XaLg, t& 8\ 'bico t6G)v ycovLcbv bQCJiisva l'6a cpaCvstaL, 
tov ccQa bfi^atog tLd^s^ivov iTtl tov Z 6ri^sCov l'6rj ctv 
(paCvoLto ij BF tfi FJ, i6tL 8% iisCtpv. i6tLV ccQa 
t6%og xoLv6g^ Sccp' o^b tdc avL6a fisyid^rj t6a cpaCvstaL. 

20 ^g'. 

yj Ei6l t^TCOL^ icp' ovg rot) Hii^atog [istatLd^siiivov tcc 
t6a ^syid^rj xal TtQbg dQd^icg Svta t(p iicoxsL^ivG} im- 
jci8c) jcoth lihv t6a^ Ttoth 8% ccvL6a cpaCvstaL. 

i6tG) t6a iLsyi^ri tdc AB^ F^ TCQbg dQd^ccg '6vta tp 

25 ijtoxsLiiLivG) iiCLTci^G). Xiyco^ otL S6tL tLg t^icog^ oi tov 
S/x/xatrog ts^ivtog tcc AB^ TA t6a KpaCvstaL. iics%sv%^G) 

1. fta'] om. V, 1/5' V, r^' m. 2 Vat. 4. tt^v] x&v v, et 
Vat., corr. m. 2. 6. /Learfoi;] corr. ex ^sltcDV m. 2 V. rjiii- 
%6%Xlov Vat., comp. v. 8. BZF] v, m. 1 Vat.; JBTZ Vm, 
m. 2 Vat. &ga] om. Vat.v. 10. tsfiviroi Vat., corr. m. 2. 



EUCLIDIS OPnCA. 101 

Est aliquis locus communis^ a quo inaequales mag- 
nitudines aequales apparent. 

esto enim maior hg quam gd, et circa hg maior 
semicirculo portio describatur et circa dg similis ei 
quae circa 6gf, et hoc est recipiens angulum aequalem 
ei qui in heg, secantes se ad inuicem portiones 
diuidantur ad z, et coniungantur zh, zg, zd, igitur 
quoniam aequales sunt qui in similibus portionibus 
anguK ad inuicem, aequales sunt et qui in hzg, gzd 
portionibus ad inuicem anguli. sub aequalibus autem 1 
angulis uisa aequalia apparent. oculo ergo posito 
super z punctum aequalis apparebit hg ei quae est gd. 
est autem maior. est ergo locus communis, a quo 
inaequales magnitudines aequales apparent. 

Sunt loci; in quibus oculo transposito aequales 1 
magnitudines et perpendiculares subiacenti plano exi- 
stentes aliquotiens quidem aequales, aliquotiens uero 
inaequales apparent. 

sint aequales quidem magnitudines ah, gd ad rectos 
existentes subiacenti ebipedo. dico, quoniam est locus, 2 
ubi oculo posito ah, gd aequales apparent. coniunga- 



2. apparerent D. 5. quae] corr. ex qui D. circa] 

contra JD. 6. portiones] portiones n D. 9. qui] mg. m. 1 D. 
21. coniungantur D. 



15. at] supra scr. V, om. Vat.v. BZF] in ras. V, 

B Vat.v. 18. (paivrito v. Icyrt] ^ctiv v. 20. ftg'] 

om. V, vj' V, vs' m. 2 Vat. 21. sIgL] litt. initial. deest in m, 
ut saepius. {lbxl^bilbvov m. 22. invjtidca] seq. Xiyta Zri 

l(rrt TLs rdnos, sed del., V. 24. Hca] om. Vm. 26. iju-- 

^Bvx^ao m, ins^s^x^onGav v. 



102 



EUCLDDIS OPTICA. 




dnb xov B inl rb ^ ij -B^, xal rstiiTjed^io Sixa 7car& 

ro E 6riiistov^ xal avrii^(o anb rov E TCQbg dQd^&g tri 

^B ii EZ. lsy(o^ ort, sav inl tfjg EZ tb Sftfta tsdf^ 

ta AB^ rj tea 
6 ^avrfistai, tlsi- ivv Ai^ 

6^(Q yag inl tfig 

EZ tb S^^a xal 

i6t(0 ro Z, xal 

7CQ06itvittit(o6av 
10 axttvsg aC AZ^ 

ZB^ ZE^ Z^, 

Z r. l6yi 6ri 

sid-sta fi ZB tfi 

Z^, aXka xal ij AB t^ F^ 'bnoxsvtav t6rj' Svo 
15 &Qa at AB^ BZ 8v6i tatg F^^ ^Z l6av sl6i, xal 

jtSQvixov6vv dQd^&g ycnvvag' t6rj aQa i6tlv i^ inb BZA 

tri vnb AZT, tk AB^ T^ aQa t6a d(pd"il6stav, 
Xiy(o 8r{^ Stv xal avv6a d^pd^jl^etav, 
fietaxsv^d^cj dii tb b^^a xal i6t(o tb if , xal ins- 
20 yvx^(o ii HE^ xal nQ067tvntit(o6av axttveg av HB^ 

HA, HT, HJ, iLSvt,(ov aQa ri HB tf^g HJ. a(prjQri- N 

^d^co ocTtb tfjg HB tfj H^ t6ri rj B@^ xal ijts^svx^^ ;•>; 

ij A@, t6rj ccQa ycjvva ij vitb B®A rfj ijtb THJ, 

&Xkoc ii i)7tb BSA tfjg i)7tb BHA ^ev^cav i6tvv^ ij ixtbg 
25 tfjg ivtdg' xal ij i)nb THA aQa tfjg i)7tb BHA i6tv 

^sv^GJV. iLsltpv aQa (pavifi6etav ij TA tfjg AB. 

Fig., quam ex V dedi, quo modo intellegenda sit, exposuit 
Weissenbom 1. c. p. 58. 

1. A\ corr. ex ^, z^ m. 2 Vat. 2. E (alt.)] supra scr. 

m. 2 V. 12. ^T/ sv^sla] in ras. V. 15. «(>«] ik^a taai codd. 

^vd] 8u6l V. AT^ Zd V. 16. Post yoi/tas del. t(S7\ &Qa 



EUCLIDIS OPTICA. 103 

tor enini ab h super d recisL\bd et dimdatur in duo 
a.eqnalia ad pxLnctam e, et protrahator a puncto e per- 
pendicnlaris es rectae dh. dico^ qnoniam^ si super e0 
ponatnr oculus^ ah, gd aeqnales apparebunt. iaceat 
enim snper ez oculns et sit z, et accidant radii az^ zh^ i 
zey zdy zg. aequalis uero recta zh rectae zd, sed ah 
ei qnae est gd posita est aequalis. duae ergo aequales 
ahy hz duabus gd, dz aequales sunt, et continentes 
angulos aequales. aequalis ergo az ei qnae est gZy 
et ad bases iacentium angulomm; quibus aequalia i( 
latera subtensa sunt tota figura. aequalis est ergo qui 
sub 6^a ei qui sub dzg, magnitudines ergo aequales 
apparent. 

dico autem, quoniam et inaequales uidebuntur. 

transeat autem oculus et sit i, et coniungatur ie, it 
et accidant radii ih, ia, ig, id. maior ergo ih quam id, 
auferatur autem ab ih ei quae est id aequaKs ht, 
et coniungatur at, aequalis ergo angulus hta angulo 
gid, sed angulus hta quam angulus hia maior est, 
quia extrinsecus scilicet intrinseco. et angulus ergo 2( 
gid angulo hia est maior. maior ergo apparebit gd 
quam ha. 

1. super d\ py/nctis del. D. 4. iaceant D. 15. con- 
iungantur D. 16. ih {alt)'] mg. m. 1 B. 18. coniungantur D. 

ietlv 7} inh BZA tjj v-jth ^ZT r] AZ rg FZ xal r&v nQhg 
rcctg pdcsaL %si\LBV(xiv ycavi&v iiXsvqolI vTtotsivovGL TKJavov avfjiia V, 
add. mg. m. 2: vq)' ccg ai Haai et: yg. ai nXBVQocl iitotBivovaiv \ 
in Vat.v post ymvlocg in textu est: ^ari &qcc iatlv ij AZ tfj TZ 
^ccl t&v itqhg tcctg §dasaL 7iSLy,sva)V yoavL&v cci TtXsvgal 'bnotsi' 
vovaLv Timvov axfjiia. 17. td] ra ydg Vat.\ sed ydg del. 19. 
drj] m, $s VVat.v. 22. &7t6] 8ri dito Vat.v. 24. yi,sl%ov v. 
25. iativ V. nsi^av iati m. 26. AB] e corr. m. 2 Vat., 
A@ V. 



104 



EUCLIDIS OPTICA. 



Ei6l x67Cov xvvig^ iv olg xov Sfiiiccxog xsd^ivxog xoc 

&vv6a [leyid^ sig xb ai>xb ^vvxsd^ivxa t6a sxaxiQO} x&v 

avv6(ov q)av7^6sxav. 
5 l6x(D yicQ iisv^(DV fi BF xfjg F^, xal tvsqI xccg BF^ 

r^ i^livxvxXva ysyQccipd^Gjeav xal TtSQlloXr^v xi^v B^. 

(ydxovv t6ri fj iv 

r© BA^ iiiiLvxv- 

xkv(p ymvva xfj iv 
10 xp BKT' 6q^ 

ydcQ iexvv sxaxiQa 

ax)x&v. t6ri aQa 

(pavvsxav 'fj B F 

xfj B^. d}6avx(Dg 
15 8h xal fi B^ xfj r^ xcbv d^iiccxcav inl xcbv BA^^ 

rZA fjfvvxvxkvcov xsviiivmv, si6v xvvsg ccQa xdjtov^ iv 

ovg xcc ccvv6a iisyid^rj di5o sig xaixb ^vvxs&ivxa t6a 

ixaxiQco xcbv ccvi6g}v cpavvsxav. 




20 EiQSvv xdnovg^ cccp^ cjv xb t6ov [liysd^og i]fiv6v cpa- 
vslxav -J) xixaQXOv fviQog fj xa&dXov iv x(p Xdy&^ iv S 
xal fi ycovva xi^vsxav. 

S6XG) l'6ov xb AZ rc3 BF^ xal tcsqI xijv AZ ys- 
yQdcpd^G) fjfvvxvxlvov y xal ysyQcc^pd^G) iv aifx^ dQd"}^ 

25 yovva fj K' xfi 8% AZ terj iexG) il BF^ xal ^qI xijv 

1. ftf'] om. V, VT}' V, v^' m.2 Vat. 2. tsd^svtog] ti%"r\xoLi v. 

3. Gvvri%"rirai v. 6. 17 BT lisl^oav Vat.v (iisl^ov v). tds] 
corr. ex rfjg V. BF] F in ras. v. 6. i^fttxvxZt v. BJ] 
m, Br VVat.v(?). 9. rg] corr. ex Trjv V. 13. qxxvriGBrai v. 

BT] BF r^ BF Y. 14. mGavrcag] «s 8' a^rtog v. 16. 

BAJ] ABJ Vat.v. 17. ra-^rdv Vat.Av. Gvvrid^ivTa Vat.Av. 



EUCLIDIS OPTICA. 105 

Sunt loci quidam, in quibus oculo posito inaequales 
inagnitudines in idem compositae aequales utrique in- 
aequalium apparebunt. 

esto enim hg maior quam gd, et circa hg ei gd 
semicirculi describantur et circa totam hd. igitur 6 
aequalis qui inh ad semicirculo angulus ei qui in hkg] 
rectus enim uterque. aequalis ergo uidebitur hg ei 
quae est hd. similiter uero hd ei quae est gd oculis 
super semicirculos ahd^ gzd iacentibus. sunt quidam 
ergo loci; in quibus inaequales magnitudines duae in ic 
idem compositae aequales utrique inaequalium apparent. 

Inuenire locos, a quibus aequalis magnitudo me- 
dietas appareat uel quarta pars uel uniuersaliter in 
proportione, in qua et angulus diuidatur. 





esto aequalis ah ei quae est gh, et circa ah de- ll 
scribatur semicirculus, et describatur in eodem rectus 
angulus h*^ ei uero quae est ah aequalis esto hg, et 

• 

8. quae (jjr.)] corr. ex qui JD. 16. et — 16. semicirculus] 
mg. m. 1 D. 

18. i-aoixiQco] ^natSQoav V. 19. ft,?]'] om. v, v&' V, vf m. 2 
Vat. 21. %ad-6Xov] nad^' S A, etVat., sedcorr. 23. ^Z(pr.)] 
AB Vat.Av, BF Vat.^m. tc5 BF] supra scr. m. 2, sed ante 
xb AZ ins., V. BF} AZ Vat.^m. AZ{Blt.)] AB Vat.Av. 

24. ijiiMviiXLOv] sequitur: iv c5 iyyByQdtpQ-fo tfiij(ia zv%6v^ sed 
del., V. iv oLijtm] iv rc5 ai)t^ in ras. v. 26. AZ\ AB Av, 
et Vat., corr. m. 2. 



106 EUCLmiS OPTICA. 

Br TtsQcysyQcc^pd^a) t^rj^a^ b dd^stat tfjg xgog tp K 
yfovCag fifiL66cav. oifxovv ii K ycovva SiTCka^Ca ietl 
tfjg ^ ycjvCag. dcjtlaeCa aQa (paCvetai i\ AZ t^ig BF 
t&v d^fidtcjv ijtl t&v AKZ^ BAF XBQiq>BQBvcbv xsc- 
5 iievcjv. 

^ 'E6tc3 6Q(hfiBv6v ti fisysd^og ro AB. XdyGj^ otc 
tb AB ixsL tdTtovg^ iv olg tov Siiiiatog ts&svtog tb 
aiftb Ttoth i]iiL6v 7tot\ 5lov Ttoth tstaQtov q^aCvstac xal 

10 xad^dXov iv ta dod^svtv Xdyo). 

TtsQiysyQaq^&G} jtSQl tijv AB xiixlog 6 AEB &6ts 
f^v AB fi'^ slvaL dLcciistQOV^ xal sClT^q^d^G} tb xsvtQOV 
tov kvkIov xal i6ta} ro f, i(p^ 0-5 xsCed^cj tb (i[ifLa^ 
xal iTts^s^ix^cj^av sid^staL aC 

16 Ar^ FB. i)jtb tfjg AFB aQa tb 

i AB ^ksnstaL. xsCed^co dij tb 
5^lia ijtl trjg tov xvxXov nsQv- 
(psQsCag xal s6tc3 tb E^ xal 
7tQ067tL7ttstcj6av dxttvsg aC EA^ 

20 EB. iTtsl ovv ii i)7tb AFB 
ycovCa trjg imb AEB iatL 8l- 
TtXri^ tb AB aQa djtb tov F 8LitXa6L0V bQataL tov 
dnb tov E. o^oCcjg xal tstaQtov fisQog ^(pd^Tl^staL^ idv 
fj y&vCa tfjg yajvCag ^ tstQaTtlfj^ xal iv tm Sod^svtL X6yco. 

26 V. 

Tcbv l'6a} td^SL (psQOfLsvc^v xal inl fLLccg TtQbg OQ&dg 
avtotg ov6rjg sid^sCag td ijtl td avtd fLSQrj TtiQata 
i%6vtc}v 7tQo6L6vta}v fiiv TtQbg rii)i/ dyofiivrjv S^d tov 

2. K] seq. ras. 1 litt. V.- 8mXccaia)v Vat.^m. 3. z/] 
m ms. V, om. Vat.Av. AZ] AB Vat.Av. 4. AKZ] 




EUCLIDIS OPTICA. 107 

circa hg describatur portio circuli; quae recipiat eius 
qui ad Aj anguU medietatein. ergo Tc angulus duplus 
est anguli e. dupla ergo apparet ab eius quae est hg 
oculis super alzh et heg periferias iacentibus. 

Esto, quae uidetur magnitudO; a&. dico, quoniam 6 
ah habet locos, in quibus oculo posito eadem aliquo- 
tiens totum, aliquotiens quarta apparet et uniuersaliter 
in data proportione. 

describatur circa ah circulus aeh, cuius circuli ah 
non sit diameter, et sumatur centrum circuli et sit g, 10 
in quo iaceat oculus, et coniungantur rectae ag, gh 
sub eo igitur qui est agh ah uidetur. iaceat autem 
oculus super circuli periferiam et sit 6, et accidant 
radii ea, eh. quoniam ergo agh angulus angulo aeh 
est duplus, ergo ab g puncto duplum eius uidetur, 15 
quod ab e. similiter quarta pars uidebitur, si angulus 
angulo uel quadruplus uel in data proportione. 

Aequali celeritate latorum et super unam ad rectos 
ipsis existentem rectam in easdem partes terminos 
habentium accedentiumque ad ductam per oculum 20 



10. diameter] -er in ras. D. 13. oculus] mg. m. 1 D. 

pariferiam D, sed corr. 15. eius] mg. m. 1 D. 

AKB Vat.Av. 6. ^H om. v, |' V, vri' m. 2 Vat. 8. re- 
^•ivxog] tiO^ritai Av, et Vat., corr. m. 2. 9. (palvBraC\ qpa* 

vslrai m. 13. icp'\ iccp' A. 15. ATB'] in ras. V; « yap 

Vat., corr. m. 2; FAB y^ AFAB k. 17. xvxZov] corr. ex 

•AivrQov m. 2 Vat. 19. nQocjtmriroi v. EA] AE y. 21. 
iariv V. 22. SiitXdeia v. 23. xat] $\ v.al A. 25. v'] 

om. V, |a' V, v%-' m. 2 Vat. 



108 EUCLIDIS OFnCA. 

Hli^atog TCccQdllrilov rg slQrj^vri sid^sia rb 7COQQ(ot£QOv 
tov Hiiiiatog tov eyyitSQOV TtQorjyst^&av Sd^Si^ ^aQak- 
Xa^dvtcov dh ro iihv TtQoriyo^^iisvov iTCaxoXovd^stv ^ tb 
dh i^axolovd^ovv TtQorjyst^d^ac. 
5 fpsQs^d^c} yuQ l6otax&g td BF^ ^Z, KA inl ficag 
^Qog dQd^dg aitotg ov6rjg sid^siag tfjg FA td iid td 
aitd iiiQrj niQata Sxovta td F, Z, A^ xal djtb rov M 
bliiiatog TtaQaXXriXog i^x^^ ''^H ^^ ^ MA^ xal iits- 
^siix^^^^'^ ^f ^J^? MZ, MA. oi)xovv ^jtQorjyoiiiisvov 

10 ^hv doxst tb BF^ i^jtaxolovd^ovv dh tb KA did tb xal 
t&v dxb tov ^iiiiatog 7tQo67tLXtov6&v dxtvvcsv tijv MF 
i%l tb r %aQf{x^ai doxstv iiaXXov t&v tcXXcov dxtCvcDV. 
ro aQa MF TtQorjysted^aL dd^sc TtQo^cdvtcov^ &g sCQrjtaL. 
TtaQaXXa^dvtcov dh tmv BF^ ^Z, KA xal d}g t&v 

15 iV^, IIP^ 2JT ysvofisvcjv %Q06m7ttstc^6av dxttvsg aC 
MN^ MU^ MZ. oixovv ro NS ^aQfjx^ccc Soxst iitl 
tb N Sid tb xal tijv MN dxttva TtaQfjx^ccL iitl ro N 
[laXXov t&v aXXcov dxtvvcov tb aQa 2JT i^tl ro T 
TtaQfjxtaL Sid ro xal tijv M2J TtaQfjx^ccL hg inl tb T 

20 iiaXXov t&v aXXcov dxtCvcov. ro il\v ccQa BF jtQO- 



1. noQQonsQov] itQoQQoitBQOv A. 3. i'jtaY,o%ov%'fi v. 6. 

gjf-] seq. ras. 1 litt. v. ^ Z] corr. ex JF m. 2 Vat. KA] 
supra scr. V. inl iii&g— 7. T, Z, A] mg. m. 2 V, mg. m. 1 
Vat.\ om. m. 6. dQd-dg] dQd-^g Vat.Av. ccvtotg] ocbtfjg 

Vat.Av. ra (pr.)] Tovs V. 7. ?;^oi;ra] ^ydi/rcov VVat.Vat.^Av. 

8. TtaQaXXriXog — MA] postea add. v. xat] in ras. V. 

ins^svx^oxsoiv] iTtsSsvx^oa in ras. V, et Vat., corr. m. 2. 9. 
ai] 7) VVat.Av. 10. doTisi — 11. ^V^aro?] postea ins. 

litt. minor. V. 11. 5^iiatog] seq. rov Sl Hiiiiatog icativcov 

7tQ067ti7ttov6&v t&v (fSQOiisvoav 7} MT tb &Qa naQaXXa^dvtoiv 
t&v BF, JZ, KA, sed del., deinde lacuna V. Post b^niat og 
del. ysvoyLSVdiv Vat.'; in V post lac. est ysvo^svatv. tcqoc- 
nmtovG&v — 16. ysvo\LSV(av] mg. V. 12. 8ov.sl v. t&v 

aXZav] om. V, 13. nQ0v.SLGhai Vat.^m. 14. t&v (alt.)] corr. 



EUCLIDIS OPTICA. 



109 



aeqnedistantem dictae rectae^ quod remotius ab oculo 
id qnod propius praecedere uidetur^ mutantibus uero 
praecedens quidem subsequi, quod uero sequitur, prae- 
cedere. 

ferantur enim aequaU celeritate hg, dZy ha super 
unam ad rectos ipsis existentem rectam ^a in easdem 
partes fines habentium g, z, a, et ab oculo quidem 



5 



K 




M. 



T 



parallelos traha- 
tur ml e\ quae 
est ga, et con- lo 
iungantur mg, 
mZy ma. igitur 
praecedens uide- 
tur hg, subse- 
quens uero Jca 16 
propter et ab 
oculo incidentium radiorum mg super g dirimari uideri 
magis aliis radiis. itaque hg praecedere uidebitur 
accedentibus, sicut dictum est. mutantibus uero hg, 
dz, Jca et sicut nx, pr, st factis accidant radii mUj 20 
mpy ms. ergo nx deduci uidetur super n propter et 
mn radium deduci super n magis aliis radiis. igitur st 
super t deducitur propter ei ms deriuari ut super t 
magis aliis radiis. igitur hg quidem praecedehs super 

6. existentes D. 7. z] z D. 17. radiorum] corr, ex 
mediorum D. dirimari] scr. deriuari; cfr. lin. 23. 20. 

mn'\ m ras. m. 1 B. 22. w] ii' JD. 24. hg] g e corr. D. 



ex xov V. 16. ST] Z r&v Vat.v. yLvoiiivav v, sed corr. 
Deinde add. iTtayioXovd-Blv Vat.^m. jtQoaitiJtrhco v. 16. 

nciQfixQ-ail itciQriXldxQ-aim. 17. iV (utrumque)] /S7 Weissenbom 
p. 60. 19. tb xat] tov v. M2\ corr. ex. M V««t. 



110 EUCLIDIS OPTICA. 

rjyoii^evov inl rov NS ysvdyLevov Sd^si sjtaxoXov&etv^ 
ro 8e AK eTcaxoXovd^ovv i%l tov 2JT yevdiievov dd^eL 
TtQorjyeL^d^ac. 

va\ 

6 ^Edv XLVGiv g)eQOfiiv(DV TcXecdvmv &vC6(p xd%ev 6vfi- 
TcaQa^peQTjxac eTtl x& aixct xal xb Sftftcc, x& fihv xp 
Sliliaxi l6oxax&9 q)eQ6iieva dd^ec e6xavav^ xdc 8e ^Qa- 
SiixeQov elg xovvavxCov q^iQe^&ai^ x& 8% d^axxov elg 
x&r ^Qorjyoviieva. 

10 q^eQi^&ca y&Q avC6a) xdxec xd: B, F, ^, 
xal PQa8iixaxa [ihv tpeQi^d^io xb B, ro 
8h r i6oxax&g x^ K S^iiaxL^ xb 8e A 
%axxov xov n a%b 8\ xov K '6yLyiaxog 
7tQ067CL7txixa)6av dxxtveg at KB^ KF^ 

15 K^, oixovv rp ii[i[iaxL 7taQaq)eQ6[ievov 
xb r e6xdvaL 86^eL^ xb 8h B 'b7tokeLit6- 
(levov elg xoi^vavxCov q^iQee&aL^ xb 8h ^, 8 '9'arrov '6jrd- 
xeLxaL xovxcov^ q^iQe^&aL 86^eL eCg xoii^iTtQo^d^ev* TtXetov 
yaQ aTtb xovxcov djto6XTfj6exaL, 

20 v^\ 

^Edv XLV(ov q)eQO[iiv(ov 8La(paCvrjxaC xl fti^ q)eQ6fLe- 
vov^ 86^eL xb fti) (peQ^iievov eCg xd ^TtL^d^ev (piQe^d^aL. 

(peQi6d^(o yccQ xd B^ /t^ [levixcj 8e xb F, xal djtb 

xov '6[ifiaxog 7tQo67tL7txix(o6av dxxtveg aC ZB^ ZF^ Zjd. 

26 ovxovv xb fiev B (peQ^fievov lyyLOv i6xaL xov F, ro 

8\ /t dnox^oQovv 7tOQQ(oxeQov' eCg xovvavxCov aQa 

(piQe^&aL 86^eL xb F. 

2. tov] t6 Vat. 4.^ vcc'] om. v, |/5' V, |' m. 2 Vat. 6. 

&viG(OV V. GVpi7t0CQ0C<pSQ7ltCcC] 6Vil7taQCC<piQ£tCCi V. 9. td] 

Bupra Bcr. m. 1 Vat. 11. ^QabvtGcta] §Qadvtoc Vat. t6 (pr.)] 




EUCLIDIS OPTICA. 111 

nx factum uidebitur sequi, at uero aJc subsequens 
super st factum uidebitur praecedere. 

Si aliquibus latis pluribus inaequali celeritate simul 
transportetur in easdem partes et oculus, quae quidem 
oculo aequaK celeritate feruntur, uidebuntur stare, 5 
tardiora uero in contrarium ferri, celeriora uero in 
praecedentia. 

ferantur enim inaequali celeritate h, g, d, et tar- 
dissime quidem feratur 6, at uero g aequali celeritate 
oculo TCy d uero celerius quam g, ab oculo uero A 10 
accidant radii M), kg, kd. itaque oculo transposito g 
stare uidetur, h uero relictum in contrarium ferri, at 
uero, quod celerius positum est eorum, ferri liidebitur 
in anteriora*, plus enim ab eis distat. 

Si aliquibus latis appareat aliquid, quod non fera- 16 
-ff jr /t ^^r, uidebitur illud non latum retror- 

sum ferri. 

ferantur enim b, d, non feratur 
autem g, et ab oculo accidant radii 
0h, ^g, zd. igitur h quidem latum pro- 20 
^ pius erit quam g, at uero d progre- 

diens longius. in contrarium uero ferri uidebitur g, 

1. at] ad D. 20. zg] zdg D. 

Tc5 V. tb 8s — 12. 6'ftftart] rc5 91 K H^iiatt. l6otocx&g '''^ ^ ^- 
14. 7tQoa7tL7tt6t(o V. 15. tco] corr. ex t6 V. 7cbqi(pbq6^8- 
vov m. 16. vitoXBiit^iLBvov] B7t6ybBvov m. 18. tovtov V, sed 
corr. 20. v^'] om. v, |y' V, ga' m. 2 Vat. 21. tpSQ6[LBvov'\ 
(puiv6\LSvov m. 22. bIq ta dTtLGd^sv] corr. ex slg ro-^j*- 

TtQoed^sv V, slg tk lybTtooe^sv v. 23. A] corr. ex P m. 2 Va 
r V. 24. 7tQoa7tL7ttEt(D V, comp. Vat. 26. B] corr. ex 

m. 2 Vat., A v. ^yyiov] corr. ex ^yysiov Y. 




i 




112 EUCLIDIS OPTICA. 

vy'. 

Tov S^^arog lyyiov xov dQO^idvov 7CQo6L6vrog 86^£c 
ro bgdiisvov rivi/Yi^d^av. 

dQded^co y&Q tb BF rov Sftftaro^ i^l ro Z xeLfiavov 

6 {mb rcbv ZB^ ZF &KrCv(ov^ 

xccl iisraTcei^&o) rb o^fia iyyiov 

rov Br xal lefrco ijtl rov ^, 

xal bQde&c^ rb a{>rb i^b r&v 

^B^ ^r &KrvV(QV. OVKOVV 

10 iisi^cav fj A ycovia rrjg Z ycj- 
vCag' ra 8% 'b^b iieC^ovog y(o- 

vCag 6Q(hfiava [isC^ova (paCvsrac, 86^6c aQa i^l^^-^^at 
ro BJT rov 8ftftarog i%l rov A Hvrog i^TteQ iutl rov Z. 

v8\ 

15 T&v t0(p rd%ei (peQOiiivcov r& ^6qqco 8oxet ^Qa- 

8vr£Qov (psQe^d^av. 

(p£QB6d'C3 y&Q C^orax&g r& B^ K^ xal &nb rov A 

Sftftarog &xrlv£g f^x%^a)6av aC AF^ A^^ AZ, ovxovv 

ro B iiL£Ctpvag lx£v r&g &7tb rov Hiifiarog &xrtvag 
20 ijy^ivag fjjt£Q ro K. ^£l^ov aQa 8id6rriiia 8v£l£ii6£rac 

xal v6r£Q0v itaQaXXd66ov rijv AZ o^ti/ 86^£c ^Qa^ii- 

r£Qov (piQ^^&ac. 

"AkkGig. 

O^Qi^d^o) y&Q 8vo 6rifi£ta r& A^ B iitl naQalkT^kcov 

26 ^vd^^c&v^ bfifia 8i ^6rc3 ro Z, &(p' o{) 7tQ06mnrirc36av 

&xrtv£g al ZA^ ZB^ ZE^ ZA. kiycn^ ori ro n6QQ(o 

ro A 8ox£t ^Qa8\>r£Q0v (piQ^^&ai roi) B. in^l y&Q 

1. vy'^ om. V, |d' V, g/5' m. 2 Vat. 2. ^yiov] corr. ex 
fyyswp Y, ut lin. 6. 4. dQ&ad^ai v. 7. tov (alt.)] corr. ex 



EUCLIDIS OPTICA. 



113 



Oculo ei; quod uidetur, propius accedente uidebitur 
res uisa augmentari. 

uideatur enim hg oculo super z iacente sub zh et 
zg radiis; et transeat oculus propius ei quod est hg 
et sit super d, et uideatur idem sub dh, dg radiis. 
igitur maior d angulus quam z, sub maiori autem 
angulo uisa maiora apparent. uidebitur ergo augmen- 
tatum hg oculo super d existente quam super z. 



6 



Eorum, quae aequali celeritate feruntur, remotiora 
uidentur tardius ferri. 10 

ferantur enim aequaK celeritate 
hy Jc, et ab a oculo radii trahantur 
agj a0, ad, igitur h maiores habet 
ab oculo quidem radios quidem 
ductos quam K minus ergo spatium it 
pertransibit h quam Tc, et posterius 
permutatis az uisum uidebitur tar- 
dius ferri. 




AUter. 

ferantur enim duo puncta a, 6 in aequidistanti- 2C 
bus rectiS; oculus uero sit z^ a quo accident radii 
za, zh, 6, zd. dicO; quod a quidem remotius 
uidetur tardius ferri quam 6. quoniam enim az, zd 



21. accident] scr. accidant. 



x6 m. 1 Vat. 8. bQ&eQ-cci v, corr. m. 1. 10. yatvlcig] om. m. 
13. Post Z add. :<^ ^gfl? V. 14. v8'^ om. v, gfi' V, Sy' 

m. 2 Vat. 17. leotaxfj Vat., corr.-m. 2. 19. iisiSovas] S 
add. m. 2 V. 20. SLsXsvGBtoci] nciQsXsvGstai m. 21. 7t€C(^ 
ocXXdaGov] naQaXXdaov V. 24. |g' V, |d' m. 2 Vat. 87. 
td] om. m. .^ 

EuclideBf edd. Heiberg et Heuge. TXl. ^ 




114 EUCLIDIS OPTICA. 

at AZ^Zjd r&v ZB^ZE ild66ova ycoviav naQiixov6i^ 
' iiatlov &Qa tb BE roi) A^ fili^srai. iocv aga r^v 

ZE &Kxlva XQO^SKpdXcoiisv i%^ 

sifd^aiag^ Sri iitl tmv l6otaxcjg 
6 q)SQOiiivcov ro ft£i/ B ijtl trig 

ZE axttvog sf xcolvd^lv i^ts- 

QSt aQa tG)v ieotax&g q)SQO- 

^iivcov td %6qqc() doxst fiQadii- 

tSQOv (piQS^d^ac. 

10 "jmcog. 

^SQiefd^co diio 6rjiista td A^ B knl TcaQaXXiqXcw 
sid^si&v t&v AA^ BE dfiaX&g' tdg t6ag aQa iv t6(p 
XQ6v(p 8isXsv6ovtau l6tci)6av 0'5v t6ai at AA^ BE^ 
xal %Q06mjttitci)6av dxttvsg dnb tov Z b^iiatog aC 
16 ZA^ Z^, ZB^ ZE. insi ovv iXdttcjv fj inb AZ^ 
trig ii^jtb BZE ycoviag^ ^'Aarrov aQa ro AA Sid^trnia 
rov BE g)av7l6stac. &6ts d6^st tb A ^QaSvtSQOV 
q^iQS^&ac, 



vs'. 



20 Tov '6[i[iatog [livovtog^ tcbv Ss bi^scov 7CaQag)SQ0- 

fiivovj td 7t6QQo tG)v 6QC3iiivc3V xataXsLTts^&ai S6^st. 

I6tc3 bQA^isva td A^ F inl sid^sc&v '6vta tcbv AB^ 

FAj bfi^a Sh i6tG} ro E^ dfp^ oi 7tQ067tt7ttitci)6av 

dxttvsg a[ EF^ EA^ EA^ EB. Xiyco^ oti ro TtQbg t& A 

26 xataXsiTts^d^ai S6i,su TtQo^sx^s^X^^^d^a) ij EA^ dxQcg 



1. ZB] BZ m. 2, AJ] corr. ex BJ Vat. (JXfWat] 

Xsinstai codd. 6. Post s lacuna Yg lin. VVat.A; s om. la- 

cuna relicta Vat.*vm. 7. &Qa] roc ndQQa dQa m. 11. JJ' 

add. V, |e' m. 2 Vat. tcc A, B] om. m. 12. AJ, BE] 



i 



EUCLIDIS OPTICA. 



115 



quam zh, ise Tninorem angulum continent, maius ergo 
be quam ad apparet. si ergo ge radium educamus in 
directo, quoniam celeritate b quidem super ze radium 
prohibet posteriorari, aequali ergo celeritate latorum 
remotiora uidentur tardius ferri. 



Aliter. 

ferantur duo puncta a^ b in aequedistantibus 

rectis ad, be, aequales aequaliter 
in aequali tempore pertransibunt. 
sint ergo aequales adj be, et acci- 10 
dant radii ab oculo za^ zb, zd, ze, 
quoniam ergo minor angulus azd 
angulo bze^ minus ergo spatium ad 
quam be apparet. quare uidebitur a 
tardius ferri. 16 




Oculo manente uisibus quoque transportatis re- 
motiora uisorum relinqui uidebuntur. 

sint uisa a^ g existentia in rectis ab^ gd, oculus 
uero sit e, a quo accidant eg, edy ea, eb. dico, quoniam 
ad a relinqui uidebitur. educatur ed, usque ubi con- 2C 



3. li] post ras. 1 litt. B. 6. Aliter] ali. B. 



AB, JE m. &Qa] om. Vat.Av. 16. ZS] om. Vm. ZE] 
ZE ycaviccg, sed ycoviotg del., V. AZJ] ZAJ m. 19. 

vs'] om. V, I?]' V, |g' m. 2 Vat. 20. nocQoctpBQOiiivaiv] -c^ 
in ras. V, TtSQKpsQOfiivoiv m. 21. td — dQaoiLivoivj mg. 

m. 1 A. v.atoL%ocldnxB6%'oti v. 22. T] in ras. V. si>^Bi&t\ 
TtocQaXXriXoav sifd^Bi&v? 24. EA] mut. in EJ m. 1 v. JBij 
supra scr. V. tc5] t6 V. 26. TLataXsinsed^oci] naXBiievstf^M 
^XQt^g — p. 116, 2. tfjg] in ras. m. 1 v. 



'^^ 



•j 




116 EUCLmiS OPTICA. 

o5 6viiPak€t tfj AB^ Tcal i6t(o ^ EB. iTCsl oiv iiei^cov 
yovia ^ 'fijro FEB tTJg imh AEB^ 
fiet^ov ccQa tb F^ Scd6trj^a tov 
AB q>aCvBtai. &6tB tov Sfiiia- 
5 tog i%l tov E iiivovtog at Sil^Big 
&g iTtl ta Aj r liiQrj TtaQa- 
q^BQd^iBvat d^attov 7CaQakkd^ov6t 
ro A i^TtBQ tb r. i^oksiTtB^d^ac 
icQa Sd^BL tb AB. 

10 v^\ 

Tct ai^avd^iBva tcbv ^iByBd^cbv S6^bl TtQO^dyB^d-av 
tp Sfifiar^. 

i6t(o 6q(6iibvov iiiysd^og tb AB^ Hii^a Sl l6t(o tb F^ 
&q)^ oi 7tQo6m7ttit(o6av dxttvBg al FA^ FB. xal rjv^i]- 

15 ^d^co tb BA Tcal i6tco tb 5z/, 3ial 7tQ067tL7ttitc3 dxtlg 
i} r^. ijtsl oiv (ibC^cov ycovCa ^ iyjtb BF^ trlg ijtb 
BFA^ fiBt^ov ccQa q>aCvBtav tb BA tov BA. td Sb 
^BC^ova iavtcbv oidfiBva iTtav^dvB^d^ac Soxov6t^ xal td 
iyyiov tov ofi^atog iKdttova q>aCvBtai. td aQa av^6- 

20 (iBva tcbv fiByBd^&v S6^BL TtQo^dyB^d^ac rp S^fiatv. 

"06a iitl rc3 ait^ Sva^ttl^atL XBttai tG}v ccxqc^v fii) 
iit' Bid^BCag t& iii6a) Si/rcov, ro Slov 6%fiiLa bt\ iihv 

XOtkoV^ bt\ S\ XVQtbv TtOLBt. 



1. cviL^oiXBl'] GviipaX^ una litt. eras. V, cvii^aXXst Vat.Av. 

tjj] e corr. V, tw Vat.Vat.^Amv. AB] B e corr. V. 3. 
lisli(ov V. 6. mg] om. m. TtaQacpSQdfisvai,] nSQicpSQOiisvaL m. 

7. itaQaXXd^ovaiv v. 8. x6 (pr.)] xov m. i)noXsinsG^aC] Xsl- 
xs0»ai m. 10. vs'] om. v, ^d'' V, |f' m. 2 Vat. 15. BJ] 



EUCLIDIS OPnCA. 117 

cuiTat ei quae est a6 et sit ei, quomam ergo maior 
est angulus geb quam aeb, maius ergo gd spatium 
quam ab apparet. quare oculo in e manente uisus 
uelut in a, ^ partes trausportati celerius permutabunt 
a quam g, relinqui igitur uidebitur ab, 6 

Augmentatae magnitudines uidebuntur oculo ap- 

propinquare. 

sit, quae uidebitur magnitudo, a6, oculus quidem 

sit g, a quo accidant radii ga, gb. et augmentetur 

ba et sit bd, et accidat radius gd, 10 
quoniam ergo maior angulus bgd 
quam bga, maius ergo apparet bd 
quam ba, maiora uero se ipsis 
uisa augeri uidentur, et eo quod 
propinquius oculo maiora apparent. 15 

quae ergo magnitudines auctae uidebuntur adduci 

oculo. 

Quaecunque in eodem spatio iacent extremis non 
in directo medio existentibus, totam figuram aliquo- 
tiens quidem concauam, aliquotiens uero conuexam 20 
faciunt. 




4. permutabunt] corr. ex permutabant D. 6. augmente, 
supra 8cr. ta m. 1, JD. 9. aumentetur D. 10. accidant D. 
12. hga] a in ras, B. 



J e corr. V, corr. ex BF m. 1 Vat.v. 17. xS] xov m. BJ\ 
corr. ex Td m. 2 Vat., F/i v. 18. ol6iLBvaJi scr. tpaiv^- 

lisvcc. Soyio^ci] om. Vat.^m, ai post lacun. VVat.v. t£\ 
rov Vat.v. 19. ^yyiov] v in ras. V. iXdttova] scr. fw(- 

iova; u. prop. V. a{)^6iisvoc] a^i^avoiisvcc m. 21. vf'] om. T, 
o' V, 1?]' m. 2 Vat. 23. 6ti\ Zxav ^, 



118 



EUCLIDIS OPTICA. 



1 yuQ ra FBJ tov Snncnog inl zov K xst- 

fidvov, xal XQoemxtitatSfiv axtivBs tcC KF, KB, K^J. 

o-itxoHv 10 SAor' tfXW"^ xoHmv S6%£i slvai. [iBtaxivBiad-a 

Sij ndXiv t6 ev tp itieip ^Qdijisvov xctl iyyiov xsieQ^a 

6 tov ^fifiatos- owxoCv tb ^BF S6^ei. xvQtbv slvai. 

'Eav terffayavov &zb rijg Ovvaipiig t&v diaiiitgmv 

XQbg 6q&us «Z'^ si&Eia, ditl Sl tavtrjs tb bfifiti re^, 

ttt xXavQul row tttQuyiavov COui ipavovvtai., xal al 
10 Sia^BtQOi S^h tSat qpavi^ffovrat. 

^ffro) tstQayavov tb ABT^, xal ^x^caaav aitoi) 

Siaydyvioi at ^B, FA, xal dv^x^^ «Qog 6^^g iab 

tov E rp ktmiSa> fiEtiajQos siid^sta 

il EZ, ig>' -^g biifia xsie&a tb Z, 
16 xal 7iQO0JCMtitmaav axzivEg al ZA, 

ZB, Zz/, Zr. iasl oh; fffij iatlv 

ij z/E t^ Sr, xoivii Si ^ EZ, xal 

aC yavtai dg&ai, ^daig Squ ii ZT 

^est t^ JZ tsti iettv, xal tav 
20 srpo? talg ^deeSt yavt&v ixElvai 

taai, 'bip' Ss al tOat xkEVQol i)XQ- 

tEtvovOtv. Cttri uQa iiitlv ^ vxb 

EZr tfiimb EZA. terj HQa ipav^- 

OEtat 15 EF t^ Ez/. 6ftoi'ws «al ^ imh AZE tf ■Oab 
25 BZE tarj iffriv. tfftj aQa <pavij6Etat ■fj AV rji BA. 

1. loii (alt.)] x6 -ra. K'] corr, es KMflou m. 2 Vat,, «ep- 
teou Y, 3. (lerffxexivTfiT&oj m. 4. rd] tm t. lYytov] t in 
raa. V. ijjiov ■KiLaO-io] foroj iyyiov Vat.v. 6. *i]'] om. v, 
oa' V, i&' m. 2 Vat. 8. rd] om. m. 10. Se\ om. Vat.v. 

gKUiiJooiitat] hio dea. Vat.'. 11. flz^to Vat.v. 12. Sut- 

■j&vioi\ -loi in ras. V. Aviijrd^aaav t. 15. Ttifoajti-Tttit v, 

cainp. Vat 18. al] om. codd, 19, M Vat.m, 21. iip' 




EUCLIDIS OPTICA. 



119 



uideantur enim ghd oculo in Tc iacente, et accidant 
radii Tcg, Tch^ Tcd. igitur tota figura concaua esse uide- 





JL. jSL 

bitur. transmoueatur uero sursum in medio uisum 
et sit propinquius oculo. igitur gid conuexum uide- 
bitur esse. 5 

Si tetragoni a contactu diametrorum ad directos 
trahatur recta, in ipsa uero oculus ponatur, latera 
tetragoni aequaUa apparent, et diametri aequales 
apparebunt. 

esto tetragonus ahgd, et protrabantur in eo dia- 10 
goni dl), ga, et protrahatur perpendicularis ab e ebi- 
pedo eleuata recta ez, in qua oculus z iaceat, et ac- 
cidant radii za, zh, zdj zg, quoniam ergo aequalis 
est de ei quae est eg, communis uero eZj et anguli 
recti, basis zg basi dz est aequalis, et qui ad bases 15 
angulorum illi sunt aequales, quibus aequalia latera 
subtenduntur. aequalis ergo angulus ezg angulo ezd. 
aequalis ergo apparebit eg ei quae est ed, similiter 

2. lig^ Tcdg D. 3. sursum] scr. mrsmn. 6. tetragoni a] 
tetragona D. 10. in eo] mg. m. 1 B. 16. dz] dg B? 

ag al HaaL] m. 2 Vat. jtXsvQai] 7t V; nXayiai Vat., corr. m. 2. 

24. Er — 25. (pavi^cstav ij] om. v. 24. AZE] des. VaK^ 

25. BJ] BZJ V. .a 



120 EUCLIDIS OPnCA. 

TtdXvv ixel ^ ^ihv FZ tfi ZB i6tiv ttfrj^ i^ ds AZ t^ 
Zz/, «AAA xal ii AB tfj Fz/, al t(f€tg &Qa tatg tQi^lv 
t6at, sl6L^ xal ycovCa yovia. t6ri aQa g)av^6etai i^ 
nlevQ& tfi TCkevQa^ &g xal at kovTCal ^kevQal t6aL 
5 (pavrj^ovtav, 

Tfig S\ iath tov Siifvatog h:l tijv 6vva<p'^v t&v 
Sca^hffcov iiilte TtQog dQd^i^g ov6rjg tp iicmiSfp ^rjte 
t6rig exatiQa t&v icjth tfjg 6vvaq)rjg ^Qhg t&g ycovvag 
tov tetQayG)VOV ayoiiivcov fwjrf t6ag ycovvag Ttovo^^^rjg 
10 iiet^ ait&v at Svd^etQOv avv6ov q)av7^6ovtav, h^ovcjg 
y^Q Sei^o^ev td^ ^v^ifiavvovta^ xad^dTteQ xal iv totg 
xihclovg. 



2. AB'] A in. ras. V. 3. siaiv v. ymvia] v.ccl ymvla v. 

4. itXBVQoi] n V. teai cpavrjaovrav} seq. spat. uac. 6 litt. v, 

om. m. 6. Ante rfjg adpon. ^ V, et in mg. haec leguntur 

initio reciso : (irjrs Ttgbg dQd^ccg .... nsS iiijrs tari '^V 

(punctis del.) kyiarsQ^ rmv icnb rfjg avvawflg TtQbg rag ycovlag 
rov rsrQay&vov icyo[LSVGiv ft^rs taag ycaviag noi . . [LSr ai)r&Vy 
al SidiisrQOL &vv6oi cpav^aovrai. d^lag yccQ $si^oiisv rcc avyir- 
paivovra, Tiad^ci^jtsQ iv rotg TivTiloLg (mg. ^aq^aXrai). In m ante 
rfjg ins. [^jai; dh i} inl ri}v avvacpfiv rfjg SuxiisrQOv iirjrs TtQbg 
dQ&ccg y rm iTtiTtiScp iiTjrs tarj r J knariQa r&v icnb rfjg avvatpfjg 
TtQbg rag ycaviag rov rsrQaymvov dcyoiisvcav iii^s taag ycaviag 
noifi iisr' avr&v %rX., quae supra e V mg. adtuli. tfjg S^ 

icitb rov biniarog] r&v iuxaTriiidroDV Vmv. 8. tarjg] tarj rg 

Vmv. rmv] ^ V. 10. &vvaov] corr. ex .... aat V, aQa 

taai V. 12. yLvvloig] -oig in ras. V, %v%Xiv,olg v. In fine: 
rsXog r&v dnrv-^&v svnXsiSov v. 



EUCLIDIS OPTICA. 121 

et angalus aze angolo hze aequalis est. aequalis 
apparebit ag ei quae est hd. rursum quoniam gz 
quidem ei quae est zh aequalis^ et az ei quae est zdy 
sed etiam ah &i quae est gd, tres ergo tribus aequales 
sunt; et angulus angulo. aequale ergo apparebit latus i 
lateri, ut et reliqua latera aequalia apparebunt. 

I 

Si uero super contactum diametrorum coniugata 
eleuata recta nec perpendicularis ebipedo spatiorum 
in contactu diametrorum nec ad rectos esse nec 
aequalis utrique a contactu ad angulos tetragoni ducta- IC 
rum nec angulos faciens aequales cum ipsis, diametri 
inaequales apparebunt. similiter enim demonstrabi- 
mus contingentia^ quemadmodum in circularibus. 



SCHOLIA 

IN 

ETJCLIDIS OPTICA. 



1. jdLtt^trjiia p. 2, 3] ^tot xata dta6tcc6si,g xal 
tag ajr' «AAiJAov dTCot^rl^SLg. 

2. ^Ev SLa^f^fiatL p. 4, 1] tovts6tL Ttatdc dLd6ta6LV. 

3. T0VtS6tLV ijtsl |xi) 6WS%Slg 7tQ067tL7ttOV6LV al 

ZipSLg^ akka xata SLa^ti^^a^ S6ovtaC tLva iv tp A^ 6 
SLa6tij^atLj TtQbg & al S^SLg oi) 7t(fo67ts6ovvtaL. 

4. ^st yd^Q t& 6Q(hiisva &7t66ta6lv tLva S^slv 7tQhg 
to 8/Ltfta' ovt(o yaQ dgad^i^^staL' 6}g st ys [iriSsiiCav S^sl 
a7t66ta6LVj o^dx ^Qad^^^^staL. 

5. MsL^(X)v av ^v tfjg FJ p. 4, 20] iidvd^avs^ SlA 10 
tL iiSL^cjv ii KA tfjg TA xaitOL i'6rj o{}6a xatd tijv 
i)7t6%^s6LV^ otav SlsI^ xal ^ EK xal fj EA Sl^ trjg Fz/. 
i7tsl 7taQdllrilog ilifjq^d^ri fj F^ tfj KA^ xal slg airticg 
ili7tS7tt(oxsv svd^sta ij KE^ iyivsto rj ixtbg ycovCa t6ri 
tfj ivtbg xal aTtsvavtCov rj {)7tb ^FE tfj iTtb AKF, 16 
SLa tbv aitbv l6yov xal ^ TtQbg t& ^ tfj 7tQbg rp A, 
l6tL S^ xal xoLvfj ycovCa i^ ^^o^ tp E' xaC sC6l diJo 
tQCyaova td FE^^ KEA tdg tQSlg ycovCag t6ag dXXfj- 
XaLg s%ovta — i\ TtQbg tip F tfj 7tQbg rp K^ ij TCQbg 
rw ^ tfj 7tQbg rw A^ xoLvij ij 7tQbg t& E — , tmv S\ 20 



1. r^. 2. V*. 3. V^ 4. V». 5. Vb. 



12. Per totum schol. E positum est pro B. ij (alt.)] 

supra scr. tfjg FJ] h. e. t&v r, d. 



126 SCHOLIA IN EUCLIDIS OPTICA. 

l^oycoviov TQcydivcjv dvdkoydv b16iv ut nsqX tdg t6ag 
ycDvCag %kBVQal Sva tov d' tov g' t&v UtOLX^Lcov. e6tai 
ovv d)g ii EA %qog tijv JF^ ovrco^ ^ EA TtQbg tijv 
AK' xal ivakkd^^ &g 'fj EJ %Qog f^v EA^ oiitcog ^ 
5 r^ ngbg ti^v KA. iisi^cjv dl ii EA trig E^' iisi^cov 
ccQa xal ri KA trjg F^. 

6. ^Tjcb nksLdvov ^scjv p. 4, 21] sl Sh imb nksv- 
6vG}v ^tIjscoVj xal ijtb nksLdvcov ycavi&v. 

7. 'Ev tp iista^ij SiaetifiiiLatL p. 6, 2] tovti6tL 
10 tSiv BF Tcal B^ STtl td i^^iTtQo^d^sv d)g TtQbg tb K 

iQXO^idvcov. 

8. Qdxovv ^Qbg tb K p. 6, 3] t&v ydQ SLa6td- 
6SCSV r} fiakkov d%o6td6SGiv 7tQOX(OQOv6&v l6taL ^sta^i> 
SLd^trj^a^ ov at d7to6td6sLg S^d tb dit* dlkijlc^v dxo- 

15 ^xc^d^rlvaL oix aijjovtaL. 

9. MsCiov Sl TtksvQd ^ J5Z p. 6, 26] ftf/gov 
si)X6y(Dg' dQd^ijv ydQ 'bnotsCvsL^ i^ Sl ZA iXdttova 
iiQ^^rig' ox) yuQ iyx(OQSt itolldg d^d^dg slvaL iv svl tQL- 
yd)vc)' Ttav ydQ tQCyovov tdg tQslg ycovCag Sv6lv dQd^atg 

20 i'6ag ix^L. 

10. Kal ii i}7tb ZBE ^. 6, 28] S^d tb slg naQ- 
akk^^lovg tiiv EB i^7ts6stv xal 7tOLfi6aL tdg ivakkd^ 
t6ag. 

11. MsC^ov aQa dg^dr^^staL p. 8, 1] S^d tbv 
25 OQOv^ otL td i)7tb iiSL^6vc3v y(ovL&v 6Q(i)^sva. 

12. f' p. 8, 5] stsQov tovto toij SsvtsQov d-so- 

8. V^. 9. V^ 10. V». 11. Y\ 



6. 
12. 


YK 


7. 
K e 


V>. 


4. 


AX] 


corr. 



SCHOLIA IN EUCLIDIS OPTICA. 127 

Qiliiatog' ixst fihv yaQ idslxvvsv^ d)g t& lyycov xsc^avcc 
&7iQV^B6tBQOv bQatav^ ivtavd^a di^ d)g fist^ov tb Syyvov. 

13. Mait,(ov d\ ycovia ^ iTtb AEB p. 8, 15] hg 
neQii%ov6a' oi) yaQ &v 7ti6y ij ET TtQbg tm A^ hg iv 
tp jJ' i^xov^ag. 5 

14. 'Ev fistscaQp p. 10, 6] ijtl tov TtQb tovtov 
d^scoQi^^atog tb fisv Sfifta ^v, £9' inCitsSov xal t& 
TtaQdkkrjka Sva^tT^^ataj ivtavd^a Sh tb S/xfta ^stsc^Qd- 
tSQOv iv ^stsGiQ^ ovtcav xal t&v Sva^trnidtcov. 

15. 'H AB p. 10, 8] fi AB oix i6tLv dxtvg^ dkkd 10 
sid^sta^ d)g djtd tvvog ^rjiisCov tov A dyo^ivrj ijtl tb 
Svd tcbv ^r^EZ ijtvTtsSov xdd^stog. b^ovcag xal ij AP 
oix dxtvg i6tiv^ dkXd x&d^stog svd^sta iitl tijv PS^ 
oi) iiijv xal TtQbg tb iTtvTtsSov xdd^stog' ij ydQ AB 
xdd^stog ijv TtQbg tb iTtoxsv^svov iTtvTtsSov. 15 

16. 'H AP &Qa i%l tiiv PS p. 10, 20] Svd tb 
Ssvxd^lv TtaQd tov nd%nov krjiiiidtvov iv totg sig td | 
^OTttvxd EvxksiSov idv djtb iistsaQov 6rj^siov iitl tb 
i^oxsi^svov iitiTtsSov xdd^stog dxd^fjj ditb S\ tov 6rj" 
^siov^ Tcad'' TtQO^pdkksv tp iitvTtiSfp ij xdd^stog^ ^X^ ^^ 
Ttdkvv xdd^stog 7tQ6g tvva svd^stav iv tp ijtvTtiSa) o%)6av^ 
xal ij dyo^ivrj djtb tov ^stsd)Q0v 6i^^siov i%^ aitijv 
xdd^stog i6tav [cfr. Pappus VI, 81]. 

17. Msi^ov aQa ycDvia p. 10, 24] iitsl dQd^oyd^vvd 
i6tvv^ at S\ pd6svg t6av^ at Sh TtksvQal avv6ov. 26 

18. ^svxtiov^ it&g ^si^cov ij i)%b SAP trjg i)7tb 
HAN. iitsl dQd^oyd^vvd i6tv td tQiycDva^ ij S^ HA 
tfjg AP iisi^cDV tQvyd)vov ydQ tov HAP fisi^cov ycjvia 



13. Vb. 14. V^ 15. V«. 16. V«. 17. V»>, Ift. y*. 



128 SCHOLIA IN EUCLIDIS OPTICA. 

'fj imh JflPA' aiipXsta ydcQ' i^ yi^Q AP TCQog ri)v PS 
itftiv difd^j oi) ^iijv xccl TtQog f^v IIB^ Zxl iirjdh TCQog 
ro iTCLTCsddv i6tvv iQ^^ifj^ Xva xccl jtQog n&6ag x&g &7Cto- 
fiivag Tcovfj dQd^Ag yovCag^ &kkk xixkvrav TCQog aitd^ 

6 ^aC i6ttv ij xkC6Lg d^sta ymvCa i{ 'bTcb BPA* aiipXsta 
&Qa ^ {)%() UPA. iisC^iov aQa ij UA tfjg AP' 'i)7cb 
y&Q tijv ^isC^ova yovCav ij ^sC^cov nksvQ& i)7CotsCvsL. 
lisC^ov 81 xal ij AN tfig AS' i^sl y&Q aC 'bnb NUA 
xal i)nb SPA dQd^aC s16lv^ iSsC%%^ri 81 ij IIA tfjg AP 

10 iisC^ov &6ts xal tb TCaQakkrjkdyQafiiiov tb 'bicb NUA 
tov i)7cb SPA ^st^ov^ xal ij tov ^isC^ovog SLd^istQog 
lisC^cjv dLd^stQOL Ss sl6L t&v TCaQakkrjkoyQ&iiiiCDV al 
NA^ SA' ij^C6ri y&Q totitcDV t& tQCycova. &6ts^ i&v 
ij PS ns6sttaL i%l tijv UN^ icpaQii66sL' l'6rj y&Q ta^vtrj' 

15 xal al PA^ AtS ivtbg ns6ovvtaL t&v AII^ AN' iXcct- 

tovsg y&Q ai)tcbv, &6ts Sl& tb xa tov a t&v 2JtoL- 

XsCcov iisC^cov l6taL ij vTcb PAS ycovCa trjg i)7cb UAN. 

otL S\ ij i)7cb nPA yovCa &iipkstd i6tLV^ ixSrjldtSQOv 

ovtcj SsL%%"ifj6staL' iTCsl tb ABP tQCycovov dQd^oyovLdv 

20 i6tLV' dQd"^ y&Q ij 7CQbg tp B' ixtbg Sh aitov ij i)7cb 
nPAj ^sC^c^v l6taL tfjg ivtbg xal a7CsvavtCov' a^ipista 
aQa. akk& xal tQLycovov tov ASN ij TCQbg t& S 
ycDvCa ^sC^cjv tfjg TCQbg rc5 N' &6ts xal ij i)7CotsCvov6a 
tijv ^sC^ova ycovCav ^isC^cov. ij ccQa AN fisC^c^v tfjg AS. 

25 19. IlQbg dQd^&g ycjvCag t6aL p. 12, 18] si yaQ tLg 
st7C0L^ hg ij HF xdd^stdg i6tL 7CQbg tijv FA^ &6a'vtG)g 
Sh ;cal ij ZB TCQbg tijv BA^ Sfjkov i6taL tb atoTCov. 



19. V2, deletum. 



19. Ante ABP del. V7t6. 



SCHOLIA IN EUCLIDIS OPTICA. 129 

ei yicQ 'li ijtb HF^ yovta d^d^j xal ^ 'bTtb BlFHl 

20. KEL^d^ca TtQog t& A y(ovCa dgd'^ [^ A^E]' 
dt^diistQog aQa ii AE. &6ts ^ iTtb EF^ yovca d^sta 
xal fi xata xoQvg)^v avtfi^ ii S\ iTtb BFE iiifiXsta 5 
xal ^ nata xoQvg)iiv aitfl r^ i)7tb HF^. &6ts i^ TtQbg 
dQd^ag ayoiisvrj tfj Tz/ r^ KF drjkadij svtbg Tts^sttac, 
Ttdktv iTtsl rj hitb BFE d^i^ksta^ d^sta ij ijtb FBE 
xal fj xatd xoQvq)^v aitfj fj {jTtb ZBA. &6ts ij TtQbg 
dQd^dg dyo^svrj tfj AB ixtbg 7ts6sttai ij SB 8rjkov6ti, 10 
ix^s^krj^d^cj^av ij ®B xal KF i%l tijv %SQLq>iQsiav^ 
xal ditb tov xsvtQOv tov xvxXov i^xd^co^av TtQbg dQd^dg 
iTtl tijv ®B xal KF ixpspkrj^svag ij AM^ AN' 
ts^vov6iv aQa tavtag SCxa xatd td M, N 6rjiista did 
tb y' tov y' t&v UtOLXsCcav. iTts^svxd^c^ ij A®^ AK. 15 
xal iitsl l6ai sl6lv a^rat" ix xivtQOv ydQ tov A' xal 
v7totsCvov6Lv dQd^dg ycovCag tdg JtQbg tp M 3cal N^ 
tb aQa d%b tfjg A® t^ov i6taL totg d%b 0M, MA^ 
6}6avta)g Sh xal tb d%b KA t6ov totg ditb KN^ NA. 
dUd ij SM tfj KN t6rj' &6ts xal ij MA tfj AN t6rj. 20 
t6aL ccQa ij ©BS^ KFU. av Sij toCvvv l6ag tavtaLg 
stSQag Siio svd^sCag dydycj^sv Svvatbv ydQ' tijv AAi 
tvxbv xal PU ts^voii^ag ^Qbg d^d^dg tijv ®BZ^ KFU 
xatd ts td Bj F 3cal T, T ^rj^sta^ xal t6(ov dtpaLQS- 
%^SL6cbv tcbv FB^ B[T]' t6aL ydQ S^d tijv t6rjv dnb 26 

20. VI 



1. BrHI FH legi non possnnt. 3. Ttsiad-G)] fort. %BtTai. 

T) AdE] euan. 13. @B] corr. ex 0J. ij] immo ai, 

sed cfr. lin. 15, 21, 23. 22. Post s^bd^sLccg del. rsiivovcag tavtag 

jtQbg ^Qd^dg. 25. T] legi non potest; idem de omnibus ualet, 

quae [] inclusi. 

Euclidea, edd. Heiberg etMenge. \IL. ^ 



130 SCHOLIA IN EUCLIDIS OPnCA. 

tov XBvtQov &ic66ra6iv' SBi%^ifi6Btat i^ 0B tfl BA t6ri 
xal ii Kr tfi rj. 

21. MBtt^ov p. 14, 15] 6g 7CBQUX0V. "Elattov 

p. 14, 16] &}Q TCBQtBxd^BVOV. 

5 22. Kal 6}g fi AB xtL p. 14, 25] l6oy6vva yd^Q ta 
EAB^ EZJ tQtyfova^ Stc ^i i)7tb E^Z t6ri i^tl ty 
imo EBA' iiiTtiTttc^xB y&Q Bid^Bta fj EB slg TtaQalki/j- 
Xovg t&g Fz/, AB' Tcal it&Uv ^ i)7tb EZJ tfi i)%b 
E\jf\B [i^tiv'^ t6ri diA ti^v aitiiv aitCav^ ij d^ itQbg 
10 t^ E xovvii xal &iig)OtiQOtg. t&v dh l6oy(ovC(ov tQi- 
yd}V(ov at mqI t&g i!6ag ycovCag jtkBVQol avcikoyov Sl& 
tb d' tov g' t&v Stoi%BCfov. hg ij AB oiv ^Qbg tijv 
BE^ il ZA TtQbg tijv JE' xal ivaXkd^^ &g ij AB 
TtQog tiiv Z^^ ij BE TtQbg ti^v JE' SitBQ iSBt SBt^au 

15 23. T&v Ar^ AA p. 18, 10] Srikovdtc axtCvcov. 

24. KotXa g)avil6BtaL p. 18, 14] tov TtoQQOtiQov 
axQov iiBtBGiQOtiQOv cpaLVO^ivov. 

25. 'Hg TtdQL^fia tovto ijtdyBLV Soxbl. 

26. TaTtBLvdtBQOv cpaCvBtaL p. 20, 1] xal y&Q %q6- 

20 %BLQOV^ OtL ta i)7tb ta7tBlVOtiQC3V dxtCvCOV 6Q(D[lBVa 

ta7tBLv6tBQa cpaCvBtaL. 

27. MbC^ovl p. 24, 13] ^ibC^ovl i)7tBQ(piQov. 

28. '76a dlltlkoLg (paCvBtaL p. 24, 20] SlA Ttldvrjv 
ti^v tfig oil}B(f}g. 

25 29. Mi%QL tov A o^^atog p. 28, 2] ag xdtcod^Bv 
tf^g dxttvog. 

21. Y^ snpra scr. 22. V^. 23. V*Vat.^ 24. Y^, 
25. V«. 26. V»>. 27. Vt>. 28. Y\ 29. V^ 



JS. ^E] jiE, 15. &%tiv(ov driXovdu Vat.^ 



SCHOLIA IN EUCLIDIS OPTICA. 131 

30. 'Slg ii ^E xrL p. 28, 10] 6l& ro 6' rov g' 
rmv 2JroLxsLcov' l6oymvLa ydcQ rct rQfycova Sia ro iv 
ratg naQaXXifiXoLg iiLnlnrsLV s^dd^stav. 

31. *'A%^Lg oh tfvfipaXst p. 28, 24] rovritfrL fiixQf^S 
av ro TciQag rot) ihl^ovg r) ro axQOv drjXadii ro A ifi- 5 
9)ai/iJ<j£rat r^ xardTcrQO) fisraxLvovfiivp' ov y&Q xar ct \^ 
TCQfDrr^v rvxbv jt^o6^okiiv rf^g oiltscag xar' i(iq>a6LV 6pa- 
^Ti^fsraL TtaQa rrjg o^il^scog iv rc3 xardmQCD ro axQOv 
rov vTpovg. 

32. 'Ev rotg KaroTtrQLXotg p. 30, 3] SlA rbv iv 10 
rotg KaromQLxotg 5qov [prop. I]. 

33. "Itfri ycovLa ij i)7tb EZB ip. 30, 25] xdd^sroi yicQ 
at EZ xal AJ. 

34. ^AXXk xal ^ {)7tb AB^ p. 30, 26] «ar<5: xopv- 
9?i)v yccQ. 16 

35. iCal ^ rQLrrj aQa p. 30, 26] Si bv k6yov 
Hvcod^sv yiyQamai, 

36. BXiTtsrai p. 34, 1] ovrcog ^ iC-^ ^AcJrroi/ 
9ai;i}<j£rafc rfjg KB (lij roiJ -^ TtQbg ry TtsQLfpsQsia So- 
xovvrog tpaCvs^d^aL^ aXk^ vjtoxdrco rov J5, xal rb E 20 
d}6avr a)g oi^x^ TtQbg r^ TtSQLfpsQsia^ aXX^ i)7tox&rco rov jd 
xal oiirojg d)g xal sid^siag &3tb roi) B Ttgbg dQd-Ag 
xarrjyfiivrjg Sl& rmv /d^ E SLfJx^ccL. aXXd Sii xal rov Z' 
xal ro Z ydcQ 'htOTidrm rov E dfpd^ritfsrai xal o{> TtQog 
rfl nsQLfpsQSLa. rbv airbv Sh r^dnov xal oatb rov Fj 26 
d}g tpaCvs^d^aL i%l (iL&g sid^sCag rflg BF ri(. B^ jd^ E^ 
Z^ H>, ®y r 6roLxsta. 

30. V^. 31. V». 32. V*». 33. V^. 34. V^. 36. V»». 
36. V*. 



5. ri] supra scr. 




132 SCHOLIA IN EUCLIDIS OPTICA. 

37. jdt&, xo 6viL^aCvBiv^ ojcsq yivexai sv&scag vno- 
xsLfiBvrig rflg vvv ov6rig nsQitpsQsiag^ vofii^sraL xal fj 
TCSQtfpiQSta svd^sta' s6ri S^k rovro ro fpaivstf&ac rag 
aTtb roi) TiivrQOv xal ravra t6ag ov6ag rijv iTcrbg (isi^cj 

6 rfjg ivrdg^ olov rijv KB rrjg 
K^ ^ o yivsrai^ sl iic^ 
sid^siag xsifSsraL i^ J5 n inl 
y&Q sid^siag tfvfipaivst rijv 
ixxsLfiivi^v olov rijv KB 

10 fisi^ova rrjg K/l slvac. si 
yccQ aXX(Dg kiysi rig ravrag 
t6ag slvai^ tfvfiPaivsL aroTtdv rr dQd^oycaviov yaQ nsi- 
fiivov rov KEB rQtythvov ro djtb rfjg pd6s(og rfjg KB 
t6ov l6rat rolg djtb rmv nlsvQmv r&v KE^ EB. ofioiiog 

16 Tcal ro dnb rfjg Kjd rolg ditb r&v KE^ E/d, Jtmg ovv s6rat 
t6ri ii K/1 rfl KB r&v djtb [rijg KE] iv t6otg td&v ov- 
rmv; fpaivsrat oiv ^ JtSQtfpiQSta si)^sta Stdrb q^aivstfd^at 
6v(iPatvov inl rf^g 7tSQtq)SQsiag^ o xal i%l rfig svd^siag, 

38. *19 OTtt^d^sv sksys dvvarbv 8sixvv6^at xal ini 
20 rfig xoiXrjg TtsgtfpsQsiag^ rovro vvv dstxvvst' olov idv 

inl roi) xivrQOv rfjg 7tSQtq)SQsiag rsd^rj rb o(i(ia^ ai Sa 
ix rov xivtQOv 'b7tors%'cb6tv Sg dxrtvsg^ (iiyt6rov filv 
fpavifi^srat ii AB sid^sta^ [rf] ro jtQorsQov dxrlg vTtixstro^ 
dsl 8h ^ iyytov rf^g AB rfjg djtforsQOV (isitcsv rfjg itQo- 
25 riQag 7tQ0xcjQ0v6rjg Ssi^sog, 

39. Kad^irov in' airijv ov6rig p. 34, 28] rfjg BT 
7tSQtq)SQsiag cjg sifd^siag voov(iivrjg. 

^ 40. ^EyxdXa6(ia p. 34, 28] rvx xoiXco^a. 

37. V^. 38. YK 39. V». 40. Y^. 



7. iTgl] insL 28. rvx] h. e. tvxbv? 



SCHOLIA IN EUCLIDIS OPTICA. 133 

41. Evd^stat yCvovrai p. 86, 3] tcsqkpsqbl&v fikv 
ovtfrjg rrjg 6mag^ 8ia Sh rag H^SQxofisvag ajcb rov qxort- 
^otnog ajtotfrdtfstg (patvs^d^at ratirag^ oiag xal iv rfl 
siy^sia^ xal slvat rotavrag. 

42. IIotrj6st ovv rofiijv xvxXov p. 36, 23] rovro 6 
sv rotg Uq)atQtxotg rov Sso8o6tov Sstxwrat [I, 1]. 

43. At FB^ B^ &Qa itp&Tcrovrat p. 38, 1] ^ r^ 
StafiirQp yd:Q rot; xvxkov itQog dQd^&g aTC^ axQag &yo- 
ybivri i(pa7Crsrat rov xvxkov^ StdfisrQog 8% ij AF rov 
FH^® xvxXov. 10 

44. 'OQd^al ccQa al TCQog rp K p. 38) 3] St& rC 
hQ^oii a[ scQog ra K; ijcsl xvxXov rov AFB^ itp- 
djcrrjrat rtg sid^sta ^ H®^ anb 8s rov xivrQOv ijtl 
rijv i%aq>iiv iTCs^svx^rj svd^sta i] BA^ ^ intt^svx^st^a 
aQa xdd^srog i^rat i%l rijv iq^aTcrofiivrjV ipO-i) aQa ij 16 
vjcb BAH. iTCsl Ss slg jrapaAAiJAovg rdg HS^ F^ 
sv%sta iviics^sv rj AB^ fj ixrbg ycovCa ^ 'bicb BKF 
l'6rj i6rl rfj ivrbg xal aicsvavrCov rfj i)7cb BAH, dQdij 
Sh il 'bnb BAH' [dQd^ij aQa] xal ^ ^Tcb BKF. 6Q%^al 
aQa al TCQbg rb K. 20 

45. 'Tnb rov ® o(ifiarog ^XiTCsrat p. 40, 10] 7C&g 
vTcb o^fiarog roi5 ® pXinsrat rb KA fiiQog rfjg ^fpaC- 
Qag; iscsl tcsqI StdfisrQOV rijv A® xvxXog 6 AASK 
yiyQaTcrat ri^vov rbv [E^F^Z x^ixlov xard rd K^ A 
[ffrj^sta]^ &7cb S^k rov \A\ 6rj(isCov [rov TciQarog] rrlg 26 
Sta^irQOv [rov AA&]K xvxkov iTcl [r& A^ K] 6rj(ista 
ilX^V^<^'^ s^b^stat at AA^ AK^ xal &7cb rov [srsQOv] 

41. Vb. 42. V^ 43. V^ 44. V^ deletmn. 45. V*. 



2. Sid] corr. ex SLta? 13. H@] H e corr. 28. n^J 
^\ AS] K@. 27. xal &7to\ corr. ex &it^ ^i, :j 






134 SCHOLIA IN EUCLIDIS OPTICA. 

Tci^axog tov & &vaxvxlovfi . . . . [af] 0A^ SK^ 

xal dQd-As ycoviag [7COLOV61] tag imo AA0^ [AK]®' 
^fttxvxAfc^ov yap' l6ti\ 8\ 8Ld(i€tQog 'fj AK xal i] AA 
tov EFAZ ixpaXXdfisvaL^ fj 0K^ 0A aQa ifpccTttovtaL 

6 tov xvxkov SLct tb 7c6QL6(ia tov l^' tov y' t&v IkoL- 
XeCGiv, &x%'BL6'Yig oiv rijg KA naQakXifiXov ov^rjg ty EZ 
ylvovtaL tit ASM^ [M^^&K tQCycova dQd^oyAvLa^ d>g 
nQoSiSaLXtaL iv tp itQh tovtov d^scoQ^^^atL. fisvo^^^rjg 
ccQa tfjg ®M {jtSQl tijv'] dQ^ijv ycavCav evd^sCag itBQL- 

10 6tQBq)6(iBvov tb tQCycovov JtOLSt tijv xcjvlxtjv ixLfpavBLav 
'fj ®A [aTtb tov] ® trig 6cpaCQag icpaTttofiivrj^ ^ 8s [A] M 
tbv xvxXov^ 06tLg i6tl pd6Lg tov xfovov. {)Jtb tcbv 
@K^ ®A HcQa dxtCvmv ofifiatog roi) ^XinstaL tb AK 
fiiQog tfjg tfcpaCQag. 

15 46. MsCtcDV ydQ ii h%b K®A p. 40, 14] itcbg ii 
JtQbg t& S ycovCa fisC^ov f^g TtQog rc5 B; insl diio 
tQCycova td BTA^ @AA tdg i)%b BTA^ ®AA l6ag 
i%ov6Lv* iv 'fj^LXVxXCoLg y&Q* i%BL Sl tb ®AA tQCycjvov 
tijv {)jtb SAA iXdttova rijg 'bnb BAF' jtsQLixstaL 

20 ydQ* XoLJtiiv aQa ffjv 'bjtb A0A ^sC^ova s%bl tfjg 'vjtb 
ABF. bfioCog xal ffjv 'bjtb ASK fisC^ova s%bl trjg 
vjtb AB^. oXrj ccQa 'fj 'bjtb ASK fisC^cov trjg 'VJtb 

rBj. 

47. naQaXXrjX6yQa(i(i6v i6tL p. 42, 13] aXXd xal 
26 Htfov rp rZ JtaQaXXrjXoyQd^^G)' l'6rj ydQ 'fj r[A] trj AB. 

48. ^EXs^v^BtaL Sh xal ijtC p. 42, 16] roi; yaQ AA 
jtsQLtftQSfpofiivov itpdilJstaL rj AB tfjg 6(paCQag^ otL xal 
tov Br x^dxXov. 

46. YK 47. V^ 48. Y\ 



1. dva-J suprsL scr. 4. 1^] h. e. ccl. 



SCHOLIA m EUCLIDIS OPTICA. 135 

49. DviifidiXXovtfL 6ii &llijlaLg p. 44, 3] 8c6tL iXdt- 
tovg eltfl p dQd^cbv al -B, F ycaviai dvo: tb xar' dvdyxrjv 
trjg afpTjg trjg dLafiitQOv tov xvxXov fisi^ovog oiitfrig. 

50. El yaQ ov 6wB^aXXov^ ^ av ita^dkXriXog fi 
{BZ^ tri rZ^ xal ro [/JE^BZ TtaQaXXrjXdyQaiiiiov^ xal 
"fl dLdfietQog t6ri \t&\ dLa^tiliiatL ' [5jcsq] oix 'bstdxsLtaL. 

51. ^Ld tC 7CQo67Cs6ovvtaL al BE^ F^; stcsI tb 
t&v d^^dtcov dLd6ti]fia fiSL^dv itftL xal jcaQdXXrjXov tfl 
dLa^itQG) trjg 6q)aLQag^ itpdiltovtaL d^k al axttvsg tfjg 
6tpaLQag xatd TtSQata SLafiitQOv xvxXov tLvbg tmv iv 1< 
trj 6q>aLQa iXdttovog xal ^a^aXXifjXov ov^rjg rp 8La- 
6trjiLatL t&v dfifidtojv^ iTCsl xal trjg dLafiitQOv tfjg 
6(paLQag iXdtftfcov avtrj i6tl xal icaQdXXrjXog^ xal oifx^ 
xatd td TciQata tfjg dLa^itQOV tflg ^tpaCqag^ at iiCL- 
^svyvv6aL tdg jcagaXXiilXovg fiiv^ fiij t6ag Si^ oix S^ovtat li 
TCaQdXXrjXoL. 6v(i7Cs6ovvtaL dqa al BE^ F^, StL Sh oix 
ifpdilfovtaL xatd td TciQata tfjg SLa^itQOV trjg 6(paCQag^ 
q)avsQ6v' slyd^ itpdilJOVtaL xatd tdniQata tr\g SLafiitQOv 
trjg 6(paCQag^ SlA ro Lrj' tov y tmv UtoLxsCcov dQd^dg 
7C0Lifj6sL ycovCag ^ ifpcactoiiivrj fistd tfjg ScaiiitQOv tf^g 2i 
6fpaCQag' at S\ dicb Siio dQd^cbv ixfiaXXdfisvac oi 6v(i- 
7Cs6ovvtaL' TCaQaXXrjXdyQa^fiov aQa i6tl tb inb t&v 
dxtCvov^ tov SLa6t7iiiatog t&v 6(i(idtC3V xal trjg SLa- 
liitQov trjg 6<paCQag 7Csqlsx6(isvov, t&v Sh TcaQaXXrjXo- 
yQd^fiojv at aTCsvamCov TcXsvQal t6aL dXXifjXaLg sl6C* 21 
teov aQa ro tav 6(i(idtG)v SLd^trjfia tb BF tfj Sca- 
[litQG) tfjg 6<paCQag' otcsq oix i>x6xsLtaL. oix iq>dil)Ovtat 
aQa xatd td TciQata trjg SLafiitQOv tfjg 6<paCQag, 

49. V»>. 50. Vb. 51. V^ 



27 ZnsQ oi}x ^jtdxetrat] supra scr. 



136 SCHOLIA IN EUCLIDIS OPTICA. 

52. ^'EXartdv i6tiv iniiXvxXiov p. 44, 8] dt& tb ks'' 
vorjd^iltG) yaQ Hii^a tb Z 7t^o6^alkov tf] lE&]^H 
6q)atQa. 

53. 'EtcsI ovv a%6 tcvog p. 46, 6] i/oiyO-i^rcj yccQ 
6 o^fia tb Z' did^ tb xa\ 

54. KvXcvdQog p. 46, 14] 6i]^eio0ai tbv xvIlvSqov 
d^d^bv [^tdfiavov. 

55. OiSatagov a^a p. 48, 1] Tcatcc ti^v imq)dvaiav 
yccQ tov xvXcvSqov ajttovtav at axfd^atai, 

10 56. x^' p. 50, 9] tb nagbv d^a^Qrj^a SaLXvvtaCy 
Sl' S)v xal tb x%' iSaix%"Yi, 

bl. Tb t6ov aQa p. 58, 9] t0ov ^lv tatg oipa^L 
tpaCvatai Si3: tb iTcb t^cav y(0VLG)v bQcc^d^aL^ ovx l6tL 
Si' ta yctQ avmtaQG) tov xmvov 6tavovvtaL, 

16 58. *'l6aL aC ycovtaL^ otL tcc iiCLitaSa totg aitotg 
iyL7caQLa%ataL tf ta^rijfta[^fci/] • £§ d)QL6iiava)v yccQ av^avcbv 
\jtaQ]aSG)xav .... ontLxov i^avax^&filyaL^ aitdg. 

59. Al rJ5, BZ avL60L p. 68, 16] Svo yccQ tQL- 
ymvd al6L tcc BFA^ BZA bQ^iiv i%ovta ycDvCav tb 
20 iikv tijv TtQbg tm F^ tb Sh f^v TtQbg tm Z, xaC i6tL 
XoLTtbv tb aTtb tf^g BA i'6ov avcc [laQog tm dnb t&v 
Br^ FA xal totg ajtb t&v BZ^ ZA. dW ^ FA fiaC- 
^cov iSaCx%"Yi trjg ZA. &6ta^ OTtaQ iXlaC%aL tijv ZA^ 
a^aL tovto 71 BZ xal S6taL [laC^cov tflg BF, 

26 60. 'EM66C3V ^hv ccQa p. 70, 1] ixaLSri yaQ tea 
aiel td djtb t&v BZ^ ZA tm djtb tcbv BK^ KA^ 

52. yb. 53. V^. 54. V^Vat.^ cum fig. 55. V^. 

56. V^ 57. Vb 58. Vb. 59. V^. 00. V^. 



15. hcci ccl y(ovioci\ postea add. 19. ra] ro. 20. T] 

eojT. ex ^. 21 et 26. rw] immo rorg, sed cn:. p. 137, 4. 



Ni 



SdHOLIA IN EUCLIDIS OPTICA. 137 

l6tL Sb^ d)g SiSBinxai^ 7] ZA ubl^cjv rijg KA^ StjIov^ 
oti fi BZ ikd66(ov itftl tijg BK' o0g) yaQ vjtBQi%Bi 
il ZA tv\g AK^ to6ovtov ilattovtai i] BZ tfig BK 
Sia td^ cog BtQrjtaL^ l6ov Bivav tb aito tobv BZ^ ZA 
tdo astb t&v BK^ KA. 

61. Mbl^(ov Sh Tcdhv p. 70, 4] [e^rat] ^bl^cjv iJ 
{)7cb BAK trjg i)7tb BAZ^ Scdti trjv vnb BAK ri BK 
{motBvvBi fiBL^cov oi^a^ hg SiSBLXtac^ trig BZ, 

62. "Hx&co ovv p. 72, 11] iitBl ri EZ itpri UQbg 
(iBV tijv Fjd TtQbg dQd^dg^ stQbg Sh tijv AB tv%ov6ag 10 
ycjvLag 7t0L0v6a^ ovx B6tL TtQbg d^d^d^g tp i)7tOKBLiiivG) 
iTtLTtiSco, 

63. 'H AM p. 72, 14] ii AM l'6ri ^iv i6tL tfj 
SLa^itQG) tov xvTclov^ oi fiijv xal SLd^BtQog^ «AA' ijto- 
tBLVov6a [iBL^ov t^rjfia ij^LXvxlLOv Sl^ tb iTtotBd^rjvaL 15 
trjv EZ l6riv vnotBd^Bt^av tfj tSN ^BL^ova tcbv ix tov 
xivtQOv, 

64. 'H NS (iB£tG)v p. 72, 19] ij yaQ EZ (ibl^cjv 
trjg ix tov xivtQOv^ ii Sl NS tfi [EZ] C^rj. [ij NS 
apa] (iBL^cov [ixatiQag^ toiv AN^ MN. 20 

65. ^H aQa TtQbg tp S ycjvLa p. 74, 1] iytBl ydQ 
il EZ 16^1 i6tl tfi SN^ ii Sh AM t6ri tfj SLafiitQp 
tov xvxkov xal tit^rjtaL SL%a xatcc tb N^ t6ri aQa 
xal ii rZ tfi AN xal ij Z/1 tfj NM. Svo Sij aC 
rZ^ ZE t6aL BL6l tfi ANj NS. xal ycavCa ij vnb ANS 26 
ycDVLcc tfi i)7tb FZE t6rj' JtQbg dQd^dg yaQ i)7t6xBLtaL 



61. V^ 62. V». 63. V^. 64. V^. 65. V*. 



7. Ti}v] corr. ex 17. 14. Ante oi) del. &lXci nal. 20. 

AN] AM{7). 25. tjj] h. e. roctg. 



138 SCHOLIA IN EUCLIDIS OPTICA. 

xal ii EZ TYi r^' pdtfLg &Qa fi EF ^d^su rfl AS Htfri^ 
Tcal a[ loiTtal y(oviac tatg XoiTtatg y(ovCaig' t6rj &Qa i^ 
i&TTo FEZ xri i)7tb ASN. 6i& xk aixk xal fj i)nh 
ZEA ttfrj xfj i)7tb NSM. okrj aQa ^ 'bnb FE^ t^rj 
6 i6xl xri iitb ASM. 

66. "EtfxaL dij xac p. 74, 8] BTtsl ij HZ t6rj i6xl 
x^ AN^ fj 8h ZE i)nsxi%'rj ttfrj xfj NO^ xal ij i)7tb HZE 
l6rj xfj imb ANO^ i6xav xal ij EH pdaig t0rj xfj OA 
xal xb XQtycovov xp XQLyihvp xal ij i)7tb HEZ l'6rj 

10 xfj i)7tb AON. iTtsl ovv sifd^SL&v x&v EZ^ ON in' 
svd^SL&v tfxad^SLtf&v ysy6va6Lv al i)7tb HZE^ ANO 
[6aL^ xal at loL7tal at i)7tb EZ®^ ONM t6aL l6ovxaL. 
xal i7tsl ij EZ^ Z® t6rj i6xl xfj ON^ NM, xal ymvCa 
ij i)7tb EZ® t6rj xfj i)7tb ONM^ pd6Lg ij E® pd6sL 

16 xfj OM t6rj xal xb XQCycjvov xp XQLymvo) xal ij i)7tb 
ZES t6rj xfj i)7tb NOM. Urj aQa ij VTtb HE@ t6rj 
xfj i)7tb AOM. 

67. 'E7tsl ovv fisC^cDv i6xlv ij 7tQ6g p. 74, 15] Sl^ 
xb xfi' xov y xcbv UxolxsCcov. i7tl xfjg avxrjg ydQ 

20 sid^sCag Svo o^ova x^Tjfiaxa xvxXcjv ox) ^v^xa^^ifj^ovxaL^ 
o^OLa Si XfLi^^axa xvxXcov xaxd xbv oqov xov aixov 
^L^Uov xd Ss%6yLSva ycovCag t6ag. oxl Sh ij 7tQbg rc5 S 
^sC^ov xfjg 7tQbg rc3 O xal TtdXLv avxrj xrjg TtQbg rp H^ 
SsLx%"ifj6sxaL Slo: xfjg SsC^SG)g xov x^' xov y' x&v IJxol- 

25 isCmv. 

68. MsyC6xrj Sh ij !S ^. 76, 6] Slcc xb Xfjfi^a xb 



66. V*. 67. V^ 68. V^. 



3. dLd] bis. r}] om. 4. FEJ] FZJ. 7. i] (alt.)] 

om. IB. Z&] Zd. xf^ e corr. 16. Ur\ (pr.)] bis. 



SCHOLIA m EUCLIDIS OPTICA. 139 

TtQO rovrov aC yaQ t6ov &'xi%ov6cci rijg Sia^stQOv 
ymvCaL t6aL el^Cv. 

69. 'TTC6Q%imELv p. 76, 9] sl yccQ l6ri^ xh S% 
'}lliLXvxXo€LShg ^xij^a 6tBvovxaL^ insQnitfy av 'fj t6ri 
avrfj, 6xsvovvtaL S\ Sl& xb i^pastxs^d^aL [r^v] a%o 6 
xov KivxQOv fiSL^dvcov oitf&v xris NS» 

70. IIsQLysyQdfpd^c} p. 78, 3] SiSsLXxaL iv x& S' 
^L^kCco rscofisxQCag tcsqI xb Sod^hv XQCycovov xvxkov -^ 
nsQLyQaifjaL, &6xs Svvaxdv i6XL rco fiovlofiivp . ^sqI 

xb KSA XQCycjvov xal hL tcsqI xb KOA xfitliiaxa 10 
xvxXcjv yQci^aL. stSQLyQaq^ivxov Sh x&v y xfirj^dxcov 
q^avsQov^ oxl fist^ov xcbv fi itfxl xb KNA r/i^fta, ro 
Sl K^A skaxxov [ft^v] aixov^ fist^ov Sh xov KOA. 
Slo: xavxa S'^ fisC^cov ij iv xp K\()]A xiiT^fiaxL ycjvCa' 
ri yicQ iv ikdxxovL xfitjfiaxL ycovCa . . fisC^cjv ' ^ Sh n^bg 16 
xp S fisC^c^v xrjg JtQbg rra N. 

71. Kal xsC6%^c3 xfl H® p. 78, 9] ijtsl yaQ xfiriiia 
x^vxkov i6xl xb KNA^ &jtb xov M tfrjfisCov JtQbg xijv 
nsQLtpiQSLav alXrj XLg ttSri X'^ MN o{>x ixfiXrjd^tfsxaL^ 
aW si t6ri xfl H0 ixpkrid^rjvaL ijtLxaxd^iitfsxaL^ ^0 ix- 20 
pirjd^T^^sxaL. 

72. ^Ejtsl ovv fisC^oiv p. 78, 18] SlA xb Xa xov y' 
xcbv UxoLxsCcjv xal Sl& xb JtQb xotjxcjv Xflfifia' d)g yicQ 
olov kflfifia iXT^q^d^rj ro . . . . \ 



69. V^. 70. V». 71. V*. 72. V»> (St.d — Zeoixsiav 

etiam A). 



8. yficoftar^', h. e. fort. (rca) yfiofter^iy. 9. 7tSQLyQdif)aC\ tcsql- 
^ v.orr. 20. &XX' st] dU' &XXri corr. ex &XXcc it&aai cci Sicc 

rovrcov y. Post KB del. ^lco. ^xjSXTj-O-^vat] in^spXTi^vccLy 
sed corr. 



e corr. 



140 SCHOLIA IN EUCLIDIS OPTICA. 

73. IlaQStfTtatfiiBvoL p. 80, 7] iftoc elg ^v (iSQog 
xaO"' SAiyv fitav dtd^stQov im(ii^x6Lg, 

74. ^Ev yd:Q t& aift^ tfiiljfiatL sl^iv p. 82, 23] vnh 
y&Q tmv aitcbv axttvcov TtSQiixetai. 

5 75. IlQog dQd^dg p. 84, 2] 6rj(i€iG)6ac^ Srt, sl stQog 
dQd^o^g i6trix€v il^ ^QX^^S^ ^Qog d^d^&g q^SQi^d^co. 

76. 'Edv dl an6 p. 84, 22] oti t6a ta tQiycova 
itavta yCvovtai td ts ijtb tfig amlvog xal t&v s^dd^si&v 
7t€Qi6x6^€va xal tovto tov TtaQdvtog Pifiliov, 

10 77. Tb ai>t6 p. 88, 3] ^toi fj AB^ ET^ ^Z' at 
ai>tal y&Q t6ai iki^fpd^rj^av. 

78. 'H(i(66ia ii i)7tb BEA p. 88, 8] Sicc tb W 
tov a tcbv UtoiX€iov' SCxa y&Q titfirjtai tb itaQaXXrik6-- 
yQafi^ov i>7tb trjg EB e^dd^eCag. 

15 79. Miyi6tov di p. 88, 15] (pav^^etai yccQ €{>qvx(0' 
QOtiQa rj JtQbg ta E ycovCa^ €i ix tov A did^etQog 
ax&eCrj n^bg tb ^iQog tov B. 

80. na6ai yaQ aC p. 90, 7] [l'6a^ y^Q td tQCycova 
[td {)]7tb tfig dxtivog [roi) 6]fifiatog xal t&v [a]jr6 

20 roi5 xivtQOv [xal tfjg AB] 7t€Qi€x6^€va. 

81. Mi6ri avdkoyov p. 92, 23] &6t€ tb vnb tCbv 
axQCJV l'6ov rc3 dnb roi) fii^ov. 

82. ^H 2J tflg B ycjvCag ^eC^cav p. 94, 11] fj TtQbg 
rc5 2J ycjvCa fieC^ov tfjg TtQbg tS B^ ijteiSfi itavtbg 

25 tQiyavov ii ixtbg ycjvCa t6ri i6tl Sv6l tatg ivtbg xal 
aTtevavtCov^ tQiycjvov Sh tov AB2J ixt6g i6tiv fj TtQbg 
tm 2J ycovCa. 



73. Vt>. 74. V^. 75. V^. 76. Y^. 77. Y^. 78. V*> 
{'di^ — 2}roixel(ov etiam A). 79. Y^. 80. V^. 81. V*. 82. V». 



\ 



SCHOLIA IN EUCLIDIS OPTICA. 141 

83. ''I6a q>avr^6Bxai p. 96, 19] xa-O*' bitoiovovv yaQ 
liSQog tijg Z^ rid^EiiBvov tov ofifiatog t6ai yovCav 
yCvovtai at TtQog ta oftftar^* t6a yaQ tQCycava xal 
o(ioLa yCvBtai tct AB®^ ©-BT, xal at fidtfBig at AS^ SF 
t6ai xal at ycavCav t6ac. 5 

84. MbC^cov aQa p. 98, 1] Stotv 'i)7tb fiBC^ovog y(o- 
vCag bQatai tijg vitb AEB tf^g {)7totBtvoiiBvrjg vTtb 
tijg AjdH 7tBQtq)BQBCag, 

85. 'ETtl tfjg EH p. 98, 2] xav xad'' btiovv^ fprj- 
6Cv^ [iBQog tfjg EH tCd^rjtat tb o^fia^ [av^L6a tpa- 10 
vi^6BtaL, 

86. Tfjg TtQbg d^d^dg p. 98, 3] tov Z di^lovdtL 
xal /1, 

87. "I6a dh q>avrj6BtaL p. 98, 22] Svvatbv y&Q inl 
tcbv BF^ r^ xal dfifpotiQov y^d^aL fiBC^ova tfiTJfiata 16 
fifiLXvxlCcjv^ atLva ov tBfiov6LV aXkrjXa^ dX^ iq>di^ovtaL 
xatd tb r 6rjfiBtov. 

88. IlQOi^yovfiBvov p. 108, 9] dvtl tov iyyvtBQev 

slvaL SoXBl t& N 6rjflBCG) VjtOL TtOQQ&tBQOV tov 2J 
67](IbCov. 20 

89. MbC^c^v ri jd ycovCa p. 112, 10] Std tb xa' 
tOV a tG)V UtOLXBCcov. 



83. Y\ 84. Y\ 86. Y\ 86. V-. 87. Y\ 88. Y\ 
89. A. 



2. Post Zz/ del. iisroc. 4. AB@, SBrj 0, e corr. 

16. Ante BF del. A. 16. ov] eras. &XX] eras. 



OPTICORUM EECENSIO 

THEONIS. 



.:^ 



^ATtoSsixirbg ta xcctd: ri^i/ oilfiv TtaQaiivd^iag ixo^L^e 
tivag nQ06€7tLkoyL^6ii6vog^ dtdtL xat' B^b^aCag yQa^^icg 
Ttav fp&g <piQ6taL. er^fiBtov dh tovtov ^eyL6tov tdg ts 
aito tcbv 0(o^dt(ov aTtOQQLTttovfisvag ^KL&g xal t&g aTtb 
5 r«i/ dvQLScjv te %al 67ta)v fpBQOfiavag avydg xo^l^bl, 
€xa6tov Sh toiitfov oi)x av iyCyv6to^ xa^ditBQ vvv 
d^scoQBttaL yLyvd^svov^ sHjteQ fti) aC dito tov iilCov 
fpeQdfiBvaL aTctlveg xatd tLvag eifd^eCag iq^eQOvto. i%C 
te tmv TtaQ* ii^tv TtvQ&v tdg ajto6t6?,lo[ievag i'q)a6xev 

10 avydg aitCag elvaL tov te qxotC^e^d^aC tLva tcbv itaQa- 
xeLfievcjv 6cj^dtcjv xal aTtOQQCnteLV 6xLdg tdg fiev t6ag 
totg 'bjtoxeL[ievoLg 6d)^a6L^ tdg Sl ^eC^ovag^ tdg S} 
iKd66ovag tcbv vTtoxeL^evcjv 6co^dtc3v. xal t6ag ^hv 
djtoQQCTtteLV 6XLdg^ o6a totg q)a)tC^ov6L jtvQOtg l6a i6tC^ 

15 tdg te i^xdtag dxttvag iitl tovtcov 6v^^aCveLv TtaQ- 
aXXr^Xovg yCyve^d^aL xal ^tjte 6vva7ttov6ag aitdg [leLovv 
tiiv 6XLdv ftijrf ^ijv i^aTtlovfievag aii^eLv^ dlV olov 

'^ i6tL ro iTtLTtQO^d^ovv ^ tOLavtrjv xat trjg 6XLdg 6vu- 
fietQiav q)vXd666LV' ikd66oveg S\ tcov 6c3fidtcov aC 6xLaC 

20 el6LV^ otav td q^cotC^ovta jtvQd fieC^ova ^* tdg ydo 
i^xdtag dxttvag 6vfi7tCjtteLv eavtatg' Slo Srj xal fieLOvv 



Tcc Tfgb r&v EvTiXsidov oittiv.cav Vpv. 1. Post o\piv add. 

6 EvyiXsldrig m. rec. V. ^xo^t^f] mut. in xoftijfi m. rec. V. 

2. dLdrC] Sl- del. m. rec. V. 4. icTCOQQiTcrov^svagj yQ. yivo- 

^Jrarff m. rec. V, dnoQQL7tro{Lsvcig p. 10. re] ys Vv. 14. Post 



Cum ea, quae ad uisum adtinent, demonstraret, 
considerationes quasdam adferebat amplius eonfirmare 
studens, omnem lucem secundum rectas lineas ferri. 
huius enim rei maximum documentum et umbras a 
corporibus iactas et radios, qui per fenestras rimasque 
feruntur, adfert. nam haec omnia ita non fierent, ut 
nunc fieri cemuntur, nisi radii, qui e sole proficiscuntur, 
secundum rectas quasdam ferrentur. et in ignibus, 
qui apud nos sunt, radios proficiscentes causas esse 
dictitabat, cur quaedam corporum obiectorum illustra- 
rentur et umbras iacerent partim corporibus propositis 
aequales, partim maiores, partim minores corporibus 
propositis. et aequales umbras ea iacere, quae ignibus 
illustrantibus aequalia essent, et in iis accidere, ut 
radii extremi paralleli fierent et neque ipsi concur- 
rentes umbram diminuerent neque uero se dififimdentes 
augerent; sed quale esset id, quod luci officeret, 
talem etiam eos umbrae mensuram seruare. minores 
uero corporibus umbrae sunt, ubi ignes illustrantes 
maiores sunt-, nam radios extremos inter se concur- 



De hac praefatione, Theonis sine dubio a discipulo per- 
scripta, u. Studien tlber Euklid p. 138—145, ubi textum Graecum 
et uersionem Germanam edidi, sed ope codicam destitutus. 

iaTi add. atg aviiPaivsLv m. rec. V. 15. avfipaivsLV^l del. m. 
rec. V. 16. yivsad^av p. 18. Post %cci add. rifv m. rec. V, 

Euclides, edd. Heiberg et Menge. \11. \^ 



146 OPTICORUM RECENSIO THEONIS. 

r&g 6XLdg. fisi^ovg di x&v 6(Ofidr(ov ai 6xiav sl^iv^ 
Srav rct (paorC^ovra jtvQ& ikd66ova fi' rdg yicQ i6%drag 
dxrtvag inl rovrcov i^ajtkov^d^at 6viL^aivBL xal [lEt^ov 
rb 6xca^6fi£vov ^SQog djtorskstv o{>Sejtor€ d' av rovro 
5 6vvi^aivBVj el [lii al dith rov itvQog g)€Q6[i€vaL dxrtveg 
iiC h^b^iiag ifpsQOvro. ixq)avi6rara Sl rovraov Ttdvroov 
rovro i^tl r&v xara6xeva6ra)g yivo(iev(ov d^eooQet^d^ai, 
^vfifiaivei. kv%vov ydcQ bit(a68rinorovv xeL[ievov et 
stQO^red^eirj rovrp jtrvxiov i%ov iitiroiiiiv keitrov itQio- 

10 viov^ &6re xal ri^v ijtvro^iiv xarc!: (ie6ov rov Kv%vov 
jtijtreLV^ rm S^ %rv%i(p rovrco xard rd ereQa (liQTj itaQa- 
re%^eiri %rv%i(yv lyyiov^ S %Q06ne6etraL ii aiyii ii Sid 
rrlg ivrofiilg (peQOiiivri^ Ttdvrcog rijv jtQ06jti7trov6av 
aiyijv r& 7trv%i^ eid^eiaig yQa[i(iatg 7teQLe%0(ievriv 

15 eiQ^^^Ofiev xal rijv iitLt,evyvvov6av r6 re [ii6ov rov 
kv%vov xal rijv itnofiiiv rov jtrv%iov xard rijv airijv 
eid^etav ov6av. 

ivaQyovg oiv ovrog rov, ort Ttav (p&g xar' evd^etav 
yQa(i(iiiv (piQerai^ xal %a6L ^rpodijAov (leraftaiveLV iitl 

20 rijv orljLV ^ltov xa\ rdg diC airrjg ix%eo(ievag dxrtvag 
xal 6(ioXoyetv xar' eid^eiag (piQe6%^aL yQa(i(idg xal rav- 
rag iv SLa6rr](ia6L^ xal SLa rovro (irjS^ rd 6Q(b(ieva 
&(ia 5Xa 6Qa6d'aL^ {)jt6(ivrj6Lv (piQav roLavrrjV nokkdxLg 
ydQ peX6vrig ^ rLvog roLOvrov eteQov 6(0(iariov ix- 

26 QL(pivrog eig rb SSa(pog (pLkorL(i6reQ6v rLveg itQ06exd- 
^•L^av rfj t,ririfi6eL xal rbv airbv r6%ov nokkdxLg i(id- 
rev6av ovSevbg ijtLStQO^d^ovvrog rc9 ^rjrov^ievo) 6(0(iarip * 



2. (poDti^oDvroi V, sed corr. 5. Gviipcclvstv p. iirji] corr. 

ex ft^ V. 9. I;u(av v, sed corr. 11. itiitxEiv] v in ras. v, 

add. m. rec. V. 12. nxv%iov] supra scr. itv^xiov m. rec. V. 

fyyBiov V, corr. m. rec. 13. Ttdvxog v, corr. m. 2. 16. yiaxd'] 



OPTICORUM RECENSIO THEONIS. 147 

rere; quare eos etiam umbras diminuere. maiores 
autem corporibus umbrae sunt, ubi ignes illustrantes 
minores sunt; in iis enim accidit, ut radii extremi 
se diflFundant et partem adumbratam maiorem efficiant. 
hoc autem nunquam accideret, nisi radii ab igne pro- 
fecti secundum rectas ferrentur. manifestissime autem 
omnium hoc in iis cemi potest, quae proprie ad eam 
rem comparantur. nam si ad lucemam quoquo modo 
collocatam adponitur tabella rimam habens tenui ser- 
rula factam, ita ut rima mediae lucemae opponatur, 
et in altera parte huic tabellae satis propinqua alia 
tabella coUocatur, in quam cadet radius, qui per rimam 
fertur, semper radium in tabellam cadentem rectis 
lineis comprehensum inueniemus et lineam, quae me- 
diam lucernam et rimam tabellae coniungit, in eadem 
recta positam.^) 

iam cum manifestum esset et omnibus constaret, 
omnem lucem secundum rectam lineam ferri, ad uisum 
radiosque ab eo effusas transiri uolebat atque concedi, 
eos secundum lineas rectas ferri et illas quidem inter 
se distantes, et ea de causa ne quae cernuntur qui- 
dem, tota simul cemi, haec admonens. saepe enim acu 
alioue eiusmodi corpusculo humi coniecto homines 
satis studiose quaerendo operam dederunt et saepe 
eundem locum perscrutati sunt, cum nihil corpusculo 

1) Debuit sic dici : line^, quae mediam lucemam rimam- 
que prioris tabellae et punctum illustratum alterius tabellae 
coniungat, semper rectam esse. 

corr. ex xai m. rec. V. 18. olv] comp. V, supra scr. olv 

m. rec. 19. iisrccpalvtov p. 20. ^|2ov] Sc^loZ m. rec. V, 

21. Ticci (pr.)] del. m. rec. V. 22. SiccatfjiiaaLv Vv. 26, 
nQOcendid^Tiaccv v. 26. XQdnov p. ifidatsvaav Y, 



148 OPTICORUM RECENSIO THEONIS. 

elra (idvroL ye v6rsQ0v impdkkovrsg ri^v oipLV rp rdjro), 
iv Stcsq fjv rb 6(o^drLOv^ sldov rijv pskovrjv. Sfjkov 
o{)Vj d)g^ ors oix s(OQ&ro rb i^SQQv^^svov^ ovd^ 6 rd- 
Tcog^ iv c5 ^v, scoQccro' &6rs rov ijcb ri^v oilfvv rov 

5 tpqrovvrog xsLfisvov rdjtov fiij ccTCavra rd (liQrj d^sco- 
Qsl6%^ai, si y&Q id^scoQStro^ xal ro ^rjrov^isvov av sco- 
Q&ro' oifx scDQccro Ss. ijci rs rcbv arsvitfivrcDV rotg 
PipXioLg 6vvi6rd(isvog lcpa6xs (irjSh rovrovg av dtJ- 
va^d^aL Tcdvra rd iv rjj 6sXiSi yQd(i(iara bQ&v. jcoXXdc 

10 yovv dvayxa^O(iivovg Sst^ai rcbv 6jcavic3g yQaq)0(iivc:>v 
yQa(i(idrcDv ft^ Siiva^d^ai Sst^ai Sca rb (lij TCQbg icdvra 
rd yQd(i(iara rd^g otlfSLg cpiQS^d^aVj dkk^ ix SLa6ri^(idrcov 
ravra^ iTcdQxsLV xa\ jcoXkd rcbv xararsray^iivcov (lij 
d^scoQstv, &6rs ix roxrcov cpavsQdv i6rL^ SLdrL oiS^ 6 

15 rdjcog rrjg 6sXiSog Zkog 6Qad^6sraL, xal iicl r&v akkcjv 
d^sa^idrcov ro a-^ro 6V(ipaivsL, &6rs ovx bQad^rj^sraL 
a(ia Ska rd 6Qd)(isva' Soxst Sh bQ&^d^aL Slu ro XLVst^d^aL 
rdg oilfSLg ijcsQPok^ rd^ovg (ii^Slv djcoXsL7Cov6ag^ rovr- 
i6rL xard 6vvsxsLav 7CaQaq>SQ0(isvag xal (lij akko(iivag, 

20 TCQbg Sh ro rfj o^sl (lij 7CQo67ci7crsLV rt stScokov 
ditb rov 6qco(isvov sig ro XLvrj^aL airijv TCQbg ro xara- 
ka^stv ro 6qc}(ilsvov icpsQSv airiag roLavrag' xa\ yaQ 
iic\ rov ^rjrov(iivov 6d)(iarog xa\ rov ra pLfikia) drsvi- 
^ovrog aTCOQiav xo(ii^c3v iXsysv si fjv ;car' siS&kc^v 

25 §(ijcra)6Lv ro bQarLxbv jcdd^og^ xa\ dicb TCavrbg 6(D(iarog 

SLrjvsx&g stSoXa dicsQQSSv^ & XLVSt r^iwv rijv aH^d^rj^Lv^ 

3. ovv] om. vp, m. rec. V. a)Q&ro V, corr. m. 1. i^- 
SQifHiivov Vp. 5. ^soQslGd^aL] -st- in ras. m. 1 V. 7. &rsvL- 
ijSavrfov V, sed corr. 8. ^(pac-nsv Vv. avvLGrdiisvog ^cpcccyis] 
del., supra scr. d^oifog (pri<sl m. rec. V. 9. TtoXXdmg V, corr. 
m. rec. 14. iart, dL&ci] mut. in iariv ori m. rec. V. 15. ^HovJ 
ai)r&v V, corr. m. rec. 17. a/ia] supra scr. m. rec. V. 18. 
Post 'bnsQpoly ras. 1 litt. v. rdxovg'] corr. ex rdxog m. 2 v. 



OPTICORUM RECENSIO THEONIS. 149 

quaesito officeret. postea uero uisu in «Bum locum 
conuerso, ubi corpusculum erat, acum conspexerunt. 
manifestum igitur est, cum res humi coniecta non 
cerneretur, tum ne locum quidem, in quo esset, cemi. 
quare non omnes partes loci sub oculis quaerentis 
positi cemuntur. si enim cemerentur, etiam res quae- 
sita cemeretur; uerum non cemebatur. et in iis 
hominibus, qui libros perlustrant, dictitabat disputa- 
tione eo conuersa, ne eos quidem omnes Ktteras in 
pagina scriptas cemere posse. saltim cum quasdam 
litterarum rariomm monstrare cogerentur, multas eos 
monstrare non posse, quia uisus non ad omnes litteras 
ferrentur, sed inter se distarent et multa eomm, quae 
subiicerentur, non cemerent. quare hinc manifestum 
est, ne paginae quidem locum totum cemi. et in 
ceteris uisis idem accidit. itaque quae cemuntur, tota 
simul non cementur. uidentur autem cemi, quia uisus 
mira celeritate mouentur nihil omittentes, hoc est 
continue transcurrentes nec desultantes. ^) 

ad demonstrandum autem, ab eo, quod cemitur, 
ad uisum imaginem quandam non peruenire, quae eum 
commoueat ad recipiendum id, quod cemitur, has 
rationes adferebat. nam et de corpusculo quaesito et 
de homine libmm perlustrante dubitationem adferens 
dicebat: si adfectus uidendi imaginibus adfluentibus 
efficeretur et ab omnibus corporibus perpetuo imagines 

1) Hucusque def. 1 explicatur. 

tovTsatLv Vv. 20. Post t6 add. fw} m. rec. V. 6^ v. 

jitTj] addidi, om. Vpv. 21. y.ivbIg&cci p. t6 (alt.)] coir. 
ex tovto m. rec. V. 22. aitlav roLccvtriv p. ^6, ^^mAn, "^^ ^ 
sed corr. &niQQSi p. "^ 



150 OPTICORUM RECENSIO THEONIS, 

xCg fi aixCMyiyvstai^ 8v ^^v ov% bga o te ^rjtaiv ti^v 
Pskdvrjv xal 6 rc3 ^i^kCfp atevC^cov jtdvta ta yQa^iiata; 
7t6teQ6v Tcote Slo^ ro ^etemQC^e^d^ai, tfi dtavoCa; akV 
ovS^v ^ittov emloyi^6^evoi ^rjtov6t xal 6ko6x^9^S oi>x 
6 ei)QC6xov6i^ Tcokkdxtg S^ d^ikovvteg itsQOig xal TceQv- 
6n:d)iievoL r^ SiavoCa eiQC6xov6i d^attov. dX}J oi icdvta 
td^ etScoXa et6xQCvetai eig tijv 5Qa6iv; xal tCg aitCa 
tov ajtoxlrjQOv^d^aL td ei6xQi>v6fieva; xal firjv tiiv 
ffii^LV ifpa6xe xatd td ^cba td ^\v tSiV ai^d^rjtrjQCcov 

10 TCQog ijcoSoxiiv cv-O-cra xate6xevaxivai^ td Se [i'^. dxoijv 
fihv ydg *Xal yev6iv xal 06q)Qrj6tv xotXa xate6xevaxev 
evtbg d)g l^cjd^ev aitatg jCQ06jcC7CteLV 6m^ata xivifi^ovta 
tdg ai^d^rl^eig tavtag. dxofj ^[lev ydQ qxov^ tcqo6' 
jcCjctov6a t6icov imtilSeLOv &(peLlev eigC^xeLv JCQog tb 

15 dvafietvai xal fiii xatd tijv 7CQ667Ct(o6LV evd^icog dico- 
TCald^^t^av tr^v te at^d^rj^LV dxCvrjtov SLaq>vldtteLV xal 
tijv i7CLq)eQ0iiivrjv 6vyxicc(' (pcovi^v, b^oCcog Se xal 
o6q)Qri6Lv' iTcl fihv ydQ yeii^ec^g tC Set xal XiyeLv; 
S^b xal iidkL6td jccog ahtaL ai ai6d"^6eLg xotXaC te xal 

20 dvtQoeLSetg xate^xevd^d^rj^av JCQog ro i^fiiveLV td tcqo6- 
nCntovta 6d)fiata TckeCovag XQovovg, xal iicl tfjg 6pa- 
6e(og otfVj eticeQ U^c^d^ev aity TCQO^iiCLicte td XLvrj6ovta 
aitijv 6(6iiataj xal ^iii aitij i^ajci6telki rt d(p' eavtrlgj 
eSeL tijv xata6xeviiv aitrjg xoCkrjv te xal evd^etov TCQbg 

25 'bjcoSoxijv t&v 7CQo6jCL7Ct6vta)v 6coiLdt(ov elvaL' vvvl Sh 
d^ecoQettaL tovto oix ovt(og ^'^^tov, dkkd iiallov 6q>aL' 
QoeLSijg ov6a d^eoQettaL ij 0Qa6Lg, 



1. yivstai p. 5. s^bQ^qanovci v, sed corr. 6. svQrjaHovat v, 

sed corr. 9. ^qxxansv Vp. ra (pr.)] t6 V. 10. xara- 

iFxsvaTiipai, v, et V, sed corr. m. rec. 11. fieV] om. v. 12. 

iSo^d-sp F, corr. m. 2. 14. ^otitTiSiov Y. 16. &va\ii]vai v, 



OPTICORUM RECENSIO THEONIS. 151 

effluerent, quae sensum nostrum adficerent, quaenam 
causa est, cur is, qui acum quaerit librumque per- 
lustrat, acum omnesque litteras non conspiciat? num 
quod cogitatione districtus sit? at etiam adtenti 
quaerunt et nihilo minus prorsus non inueniunt, saepe 
uero cum aliis coUoquentes et cogitatione diducti 
celerius inueniunt. an non omnes imagines in uisum 
penetrant? at quaenam causa est, cur eae, quae pene- 
trant, seligant^ir? praeterea dictitabat, naturam in 
animalibus alia instrumentorum, quibus sentiant, ad 
recipiendum apta comparasse, alia non apta. nam in- 
strumenta audiendi, sapiendi, odorandi introrsus caua 
comparauit, ut extrinsecus ad ea corpuscula adcidant 
ad sensus illos mouendos. nam uox ad aurem ad- 
cidens locum aptum inuenire debebat, ut maneret neue 
in adcidendo statim repulsa sensum immotum relin- 
queret uocemque adlatam confunderet. et de sensu 
odorandi similiter. nam de sapiendo quid opus est 
uel uerbum facere? quare etiam haec maxime instru- 
menta sentiendi caua et cauemis similia comparata 
sunt, ut corpora adcidentia diutius manerent. itaque 
uisum quoque, si corpora, quae eum mouerent, extrin- 
secus adciderent nec ipse ex se aliquid emitteret, 
cauum comparatum esse necesse erat et ad corpora 
adcidentia recipienda aptum. nonc uero hoc non ita 
esse adparet, sed potius sphaerae similis oculus esse 
cernitur. 



sed corr. 19. ccl] ins. m. rec. V. 21. nlsiova %q6vov m. 

rec. V. 22. rinsQ v, sed corr. nqoaimnxBv V. 23. i^T 
aniatsXXsv Yy. Post kavti^g add. ngbg icvtiXri^^^iv t&v 6qcsv&ip 
m. rec. V. 26. ^x^'^ ^y ^ed corr. <s<pttt^o%v$%V^ ^ ^ ^^^ "^«nDU 



152 OPTICORUM RECENSIO THEONIS. 

TCQog o5v ro jtL6tbv alvai xcctct tb TCaQOV tb ax- 

^ ttvag Bivai t&g ixxBOfievag xal xivovfSag tb bQatixbv 
ndd^og iQXOiivtcog i86x£i eiQYi^^^aL^ JtQbg 81 tb to^g iv 
tp ait^ iiciicidoi tatg oiI;s6l xet^ivag iceQifpeQeCag 

6 sid-eCag (paCve^d^ai ikeye tdSe' Sv6ti ii iv rp a^»Tp 
iiciiciSo} xei(iivri oilfLg ^tiVLOvv d^ecjQrjtp toiavtrj i6tlv 
&6t£ ftiJTf i^riXotiQa ^lvai, tov d^^cjQOVfiivov ftijTa 
tajC£cvotiQa' tb yicQ iv t& avtdi ijcmiScD x£tc!%^au tovt^ 
i6tiv, £l ovv ovt£ ta7C£ivotiQa ovt£ iilfrjXotiQa iiStlv 

10 r] ii^ig trjg iv t^ imjciScD y£yQa(i[iivrig n£Qi^£Q£Cag^ 
fybj)^ tot6S£ ^hv totg fiiQ£6LV ir^rikotiQag 7CQ06^dXk£L 
dxttvag tot6S£ Sl tajC£tvotiQag^ akl&, 7Ca6i totg fiiQ£6v 
trlg iC£Qtq)£Q£Cag t6ag ticg Stk tov iicticiSov (p£QO[iivag 
dxttvag 7CQ06^dkk£t &6t£ ti^v airciiv yCyv^^d^at altCav 

16 Tov t£ tb ijcCjC£Sov ^ifd^^Cag g)avta6Cav dicoktTC^tv xal 

.' tijv iv tp iicticiSco y^yQaii^ivrjv 7C£Qtq)iQ£tav. xal yaQ 
tb ijcC7C£Sov tb iic^ ^ifd^^Cag x£C^£vov trj ofl>£t avtb fihv 
dd^^djQrjt^v i6tl Std ro ^ij 7CQ067cCict£tv aitp iirjS^iiCav 
t&v dnb trjg o^£CDg ixxeofiivcov dxtCvcov^ ro Sh TciQag 

20 avrov d^ec^Qettat^ Sjcbq i6tlv i^ 7CeQtq)iQeta, kiyet Sh 
[Std^ f^v TCQbg tfj o^et xet^ivrjv yQa^^iTlv^ ijttg totg 

^^ kotjcotg tov imTciSov ^iQe6tv ijctTCQo^d^ov^a dd^ed^Qritov 
TCOtet tb ijcCjceSov, fj Sh a^dtij aitCa ij jcbqI tov int- 
TciSov tov i%^ exfd^eCag xetfiivov rp o^fiatt jcotet eid^eCag 

25 djcoStS6vat (pavta6Cav xal t&v 7ceQt(peQetGiv t&v iv rc3 
aift^ imjciSG) xetfiivcov rra ii^^att, (paCve6%^at S\ tb 
fihv ^et^ov^ otav TckeCoveg oipetg iictpdkkco^tVj ro Sh t6ov^ 

1. x6 (pr.)] xovxo V, corr. m. rec. bIvuC] in ras. m. rec. V. 

4. rars] corr. ex xag m. rec. V. 6. l^XsyBv V, v eras. v. i^ 

om. pv. wbxm] bis p, et v, sed corr. 8. xccTCSLvax^Qa V, 

et V, sed corr. 9. vipdoxsQa v, sed corr. icxlv] -Lv in ras. 

m. 1 V. 14. ylvead^at p. 17, btpri v. 20. nsQKpsQSta'] 



i 



OPTICORUM RECENSIO THEONIS. 153 

ad confirmandum igitur in praesenti, radios effundi 
et adfectum cemendi mouere, satis dictum esse uide- 
batur, ad demonstrandum autem, arcus in eodem plano 
positos, in quo oculos, rectas adparere^), haec dicebat: 
oculum in eodem plano positum cum quolibet uiso 
eius modi esse, qui neque altior uiso neque demissior 
esset', hoc ipsum enim esse in eodem plano positum 
esse. iam si oculus neque altior neque demissior arcu 
in plano descripto est, non his partibus altiores, 
illis autem demissiores radios adiicit, sed omnibus 
partibus arcus aequales radios, qui per planum feruntur, 
adiicit, ita ut eadem sit causa, cur planum rectae 
imaginem relinquat, et cur arcus in plano descriptus 
idem efficiat. etenim plahum ad oculum in directo 
positum ipsum quidem non cernitur, quia nullus radio- 
rum ex oculo effusorum ad id adcidit, uerum terminus 
eius cemitur, hoc est ambitus (lineam dicit ad oculum 
positam, quae reliquis partibus plani officiens prohibet, 
ne planum cematur).^) eadem autem causa, quae de 
plano ad oculum in directo posito ualet, etiam efficit, 
ut ex arcubus in eodem plano positis, in quo oculus 
est, imago rectae proueniat. aliud autem maius adparere, 
ubi plures radii^) adcidant, aliud aequale, ubi aequales, 

1) Prop. 22 explicatur et confirmatur. 

2) Haec uerba discipulus de suo addidit ad explicandum 
uocabulum TtSQKpSQSicc. 

3) Debuit dici iLsiSovss ycoviai. ceterum quae sequuntur a 
uocabulo (paivsad^ai lin. 26, male cum praecedentibus cohae- 
rent nec hic locum habere uidentur. nisi lacuna maior est, 
discipulus uerba Theonis parum intellexit. 

sv&sta yQafHLi/j V. 21. dta] deleo. Si^ v. %eniivri v. 28. 
ig(alt.)] in ras. V. 26. iiTCodMvai] &noSod^fjvai? Post %al add. 
nsgi m. rec. V. 27. nXsiovoq V, corr. m. t^o., iitx^^ii^fiww ^ 



154 OPTICORUM RECENSIO THEONIS. 

orav tdui^ th 8\ iXa66ov^ Ztav ilcc66ov£s yiyvc3Vtai t&v 
o^ecDV olov ymvCai tiv\g TtQbg t& oftftan. 

"Oqoi. 

a\ ^TTtoxsL^d^o) tag &7cb tov ^iifiatog otl^sig xat' 
5 evd^slag yqa^^icg q^SQS^d^at did^trjfid ti TCOioii^ag ht^ 
dkknllov, 

p\ xal tb ^sv vjtb t&v btlfsc^v 7Csqu%6^svov ^XVl^ 
slvai x&vov f^v xoQVfpijv iihv i%ovta TCgbg tp b^^atCj 
t^v dh pd^LV TCQbg totg TtSQadc tcbv bQcofisvcov. 

10 y\ xal bQadd^ai ^h/ tavta^ TtQbg ct av a[ 'dil^Sig 
7CQ0(SicCictG)6iv^ fiil bQa^d^ac ds^ TCQbg & &v ft^ jcqo6- 
7cC7Cta)0vv a[ Sjl^sig. 

d\ xal td fisv VTcb fisC^ovog yovCag 6QG)fisva [isC- 
%ova (paCvs^d^ai^ td 8% {mb ikd66ovog skd66ova^ t(Sa S^ 
15 td i)7cb l'6a}V ycovv&v 6Qd)^sva. 

s\ xal td filv iyicb fistscjQOtSQCov dxtCvcav bQ&iisva 
listsa)Q6tsQa (paCvsO^^ai^ td 8\ i)icb taicsivotsQGDv ta- 
%siv6tsQa, 

g'. xal biioCcsg td filv i)7cb 8s^tG)tiQG)v dxtCva)v 
20 &Qd)^sva 8s^Ld)tSQa (paCvsfS^^av^ td 8\ i)jcb dQL^tsQcots- 

QCJV dQL6tSQd)tSQa. 

^\ td 8s i)7cb 7cXsl6vov ycDvvov 6Qd)[isva dxQLfis6ts- 
Qov (paCvs^d^aL. 



2. oiav V, sed corr. 3. ^qol] mg. m. 1 V; oqol dnnyiol 

ins. m. 2 p; ivtsvd^sv ol Sqol r&v Ei^yiXsldov dittVK&v mg. m. 

rec. V. numeros om. Vpv. 8. tai] corr. ex t6 m. 2 v. 9. 

nsgaavv Vv. 10. ai otpsis] ras. 3 litt. v. 11. 7tQoanint(o- 

iFip (pr.)] -Tetci)- supra scr. m. 1 v •, praeterea supra add. jJ ; 



OPTICORUM RECENSIO THEONIS. 155 

aliud minus, ubi minores quasi anguli quidam radio- 
rum ad oculum existunt. 



Definitiones. 

1. Supponamus, radios ex oculo secundum rectas 
lineas ferri inter se distantes. 

2. et figuram radiis comprehensam conum esse, 
qui uerticem ad oculum, basim autem ad terminos 
uisorum habeat. 

3. et ea cerni, ad quae radii adcidant, non cemi 
autem, ad quae radii non adcidant. 

4. et ea, quae a maiore angulo cernantur, maiora 
adparere, minora autem, quae a minore, aequalia autem, 
quae ab aequalibus angulis cemantur. 

5. et ea, quae sublimioribus radiis cernantur, sub- 
limiora adparere, quae autem |a demissioribus, de- 
missiora. 

6. et similiter ea, quae a dexterioribus radiis cer- 
nantur, dexteriora adparere, quae autem a sinistriori- 
bus, sinistriora. 

7. ea autem, quae a pluribus angulis^) cemantur, 
clarius adparere. 



1) Exspectaueris oif^soov. 



seq. ai ^tpsvg^; itQoaitlnxaiav p. TtQoanintoDaiv (alt.)] sr^otf- 

nlatv V, corr. m. 2. 14. Si (pr.)] d' p. 19. Ss^toTiQotv "V 
20. &QLat6Q0tiQa)v V. 22. Si] d' p; >tal hi ta int6 il 

rec. V. ^ 



156 OPTICORUM RECENSIO THEONIS. 



a . 



OvS^v T&v 6Q(X)fi£V(ov Sifia olov bQccrav. 

b6x€o y&Q 6Qd)ii€v6v ri> ro j4^j Hfifia Sh s6rG) rb Bj 

a(p o\) 7tQo67tcjtrsr(o6av oipsf^g al BA^ BF^ BK^ B^. 

5 oifxovv ijtsl iv Sta^rTliiarL tpSQOvrai al jtQo67ti7trov6ai, 

SipSLg^ ovx ctv TtQO^jtiJtroisv 6vvs%slg %Qog ro A^, 

&6rs ysvoiro av xal xara ro AA Sta^rT^^ara^ JtQog 

a aC ^dilfsig (rd stQo6jts6ovvraL. o^bx aQa d^pd-i^^srai a^La 

Zkov ro AA. SoKst Sh dQ&^d^at Sfia r&v 8il;s(ov raxi) 

10 7taQaq)SQO[isv(ov, 

Tmv l'6(ov ^sysd^&v iv Sta^r^^fiari xsl(isv(ov ra 

iyyiov xsifisva dxQifis^rsQov bQarau 

s6r(o ^fi^a fihv rb B^ bQG)fisvov Sh ro FA xal rb 
15 KA' XQij Ss voslv ai>rd l'6a xal jtaQdllrjla^ lyycov Ss 

l6r(o ro FA' xal 7tQ067ti7trsr(o6av 6^SLg 

d)g af Br^ BA^ BK^ BA. oi yuQ av 

si^TtOL^sv^ c3g al ditb rov B H^iiarog 

jtQbg ro KA 7tQo67tLmov6aL bjl^SLg [d)g^ 
20 SLa rc)V -T, A 6rjfiSL(ov iksv6ovraL. 

i) ydQ av rQLy(ovov rov BAAKFB rj 

KA fiSL^cov &v ^v rrig FA' vno- 

xsLraL S% xal t^rj. ovxovv ro FA i^tb jtXsLdvcov aipscav 

bQaraL fjnsQ ro KA. dxQL^s6rsQ0v ccQa (pavifi6sraL ro 
25 FA rov KA. 

y ' 

"Exa6rov rcbv bQca^svcov i^SL rt ftijxog dito^rT^fiarog^ 
ov ysvdfisvov oixsrL oQaraL. 

6. TtQOGTCiTcretsv V. 7. xat] del. m. rec. V. 12. Slcc- 

erriy^aL m. rec. V. Post TiSL^svoiv add. ScvlaoLg m. rec. V. 

13. fyySLov V, corr. m. rec. 14. dgday^svav m. rec. V. 15. 

fyyeiov F, corr. m. rec. 18. al^ om. p. 19. rd] corr. ex 





OPTICORUM RECENSIO THEONIS. 157 

1. 

Nihil eorum, quae cemuntur, simul totum cemitur. 

cematur enim AA ^ oculum autem sit 5, a quo 

radii adcidant BAy BF, BK^ Bjd, itaque quoniam 

radii adcidentes in distantia feruntur, 

continui non adcident ad A^ [def. 1]. 

quare in A^ quoque interualla orien- 

tur, ad quae radii non adcident. 

ergo A^ simul totum non cemetur. 

uidetur autem simul cemi, quia radii 

celeriter transcurmnt. 

2. 

Aequalium magnitudinum in distantia positamm 
eae, quae propius positae sunt, clarius cemuntur. 

oculus sit B, cemantur autem FAj KA. oportet 
autem ea aequalia et parallela fingere, et propius sit 
FA. et radii adcidant ut BFy BA, BK, BA. neque 
enim contendere possumus, radios a B oculo ad KA 
adcidentes per puncta F, A ituros esse. ita enim in 
triangulo BAAKFB recta KA maior esset recta FA. 
at supposuimus, eas aequales esse. itaque FA a pluri- 
bus radiis cemitur quam KA. ergo [def. 7] FA 
clarius adparet quam KA, 

3. 
Omnia, quae cernuntur, longitudinem quandam 
distantiae habent, ubi cum posita sunt, non iam cer- 
nuntur. 

xriv m. rec. V, xa v. a>s] del. m. rec. V. 21. BJAKF -p, 
et V, sed post F ras. 1 litt. 22. &v] del. m. rec. V. %1. 
ri\ X seq. ras. 1 litt. v. 28. ov] iv & m. rec. V. ^MIO^ 

fifvov p et corr. m. rec. in ysvo^ievov Vv. j| 



158 



OPTICORUM RECENSIO THEONIS, 




s6ta) y&Q Hfifia fihv rb B^ dQo^iievov Ss ro F^. 
q^r^fil tfij, ort ro F^ iv ttvi ocTCodtilfiati y£v6^£vov 
oiyxiti iQad^il^staL, ysyevi^^d^G} y&Q 
ro r^ iv ta i^sta^v SiadtiliiatL t&v 
6 8^€G}i/, i(p* 0-5 ro K, oixovv JtQbg 
ro K oi)S£[i(a t&v dxb tov B '6i^£(ov 
7tQ067t£6£ttaL [itQbg 8 Si y£ aC '6il}£ig 
ox) JtQOfS7tCittov6iv^ ix£tvo oix iQatai]. 
€xa6tov aQa tcbv bQcoiiivov i%£i tv [17}- 
10 xog icjto^tri^atog^ oi y£v6ii£vov o^Tciti 
bQatai, 

Tcbv l'6(ov Sia6tYiiLat(Qv iitl trig a^dtijg ^id^^iag 
'6vt(xiv tk ix 7tl£Covog a7to6ti/jfiatog 6Q(0fi£va iXdttco 

15 (paCv£tav. 

i6t(D yScQ t6a tcc BF^ Tz/, ^Z, ofi^a Sl ro K^ 
&(p* oi jtQ06jtLJttit(o6av (itlf£cg aC KB^ KF^ K^^ KZ' 
rj Sh KB TtQbg dQd^ag £6t(o t^ BZ. ijt£l ovv iv dQd^o- 
ycjvCp tQiy(bv(p t^ KBZ t6ai £l6lv aC BF^ F^^ AZ^ 

20 [i£C^c3V i6tlv fi ^hv E yc3vCa trig H ycovCag^ f^ Sh H 
ycovCa trjg ® ycovCag, ^£t^ov &Qa (paCv£tai ro ft^v BF 
rov rj, ro Sl FJ tov JZ, 

£\ 

Tdc t6a [i^yid^rj avi6ov Si£6tri7i6ta avi6a (paCv£tai^ 
25 xal iL£ti/ov al£l ro iyyiov tov '^^iiiatog x^C^i^vov, 



1. rj — 2. r(J] add. m. 2 v. 2. qpTjjitl diQ Xkym v. 7. 

TT^ds — 8. b(^&xai\ om. p. 8. iyislvca v, sed corr. 10. ysvo- 

lisvov V, et V, sed corr. m. rec. 13. diaaTTiiidtoDv] iLsys&mv 

m. rec. V. IQ. Post taa add. iisysd-ri m. rec. V. 22. Post 



OPTICORUM RECENSIO THEONIS. 



15 



sit enim oculus B, cernatur autem Fz/. dico igi- 
tur, JTz/ in quadam distantia positum non iam cemi. 
ponatur enim F^ in distantia [def. 1] radiorum uelut K, 
itaque ad K nullus radius a B adcidet. uerum ad 
quod radii non adcidunt, id non cemitur [def. 3]. 
ergo onmia, quae cernuntur, longitudinem quandam 
distantiae habent, ubi cum posita sunt, non iam cer- 
nuntur. 

4. 

Longitudinum aequalium in eadem recta positarum, 
quae e distantia maiore cemuntur, minores adparent. 

sint enim aequales Br, JT^, 
z/Z, oculus autem sit K, a 
quo adcidant radii KB, KF^ 
KJ^ KZ] KB autem ad 5Z 
perpendicularis sit. iam quo- 
niam in triangulo rectangulo 
KBZ aequales sunt BF, Fjd, 
JZ, erit LJB:>H, LH>®. 
ergo BF maius adparet quam 
JTz/, r^ autem maius quam 
^Z. 




0. 



Magnitudines aequales inaequaliter distantes in- 
aequales adparent, et semper maior, quae oculo pro- 
pior est. 



jd Z add. z&v HiQoc ^sysd^&v inl rfjg a{>rfjg sv&siag Hvtatv tic ht 
nXsiovog &noarifiiiarog dQ^fisvcc iXdrroa (paivsrai m. rec. V. %b, 
J^ysiov V. .4j 




160 OPTICORUM RECENSIO THEONIS. 

s6tG) y^Q l'6ov rb F/l r© KA^ 
^[ifia Sh ifStcn tb B^ ag?' o5 %qo6' 
7tmtdt(o6av S^atg a[ BjJ^ BA^ BK^ 
B r. oixovv ro F^ ijcb iiei^ovog y(o- 
6 viag OQatai i^TtSQ tb KA' [let^ov aga 
fpaivetai tb Fjd tov KA. 

Tct TCaQaXXrjXa tcbv Sia^trnLdtcov ii, aJto6t7]fiatog 

bQibfisva &vi6o%Katri (paCvstav. 

10 i6tco yccQ tb BF tm ^Z jtaQccllrjlov did^tYjiia^ 

S^ifia ds i6tc3 tb K, XiycD^ Zti tk BF^ ^Z &vi607tkatri 

(paCvstaiy xal fist^ov &sl tb syyiov dLd6tri[ia tov tcoq- 

Q(OtSQ0V. 

7tQ06jtcjttsta)6av docttvsg at KSj KA^ KII^ KN^ 
15 KB^ K^^ xal STts^svxd^co^av s^b^tstai al SA^ TIN^ BJ. 
insl ohv ^sC^a)v i6tlv 'fj ijtb SKA ycjvCa tijg vTtb 
IIKN ycovCag^ [isC^cjv aQa (paCvstai xal fi SA svd^sta 
tijg IIN. Sia ta aita S^ xal ri IIN svd^sta ^sC^cov 
(paCvstai trig B^ sifd^sCag, oixiti oiv d^pd^ij^stai JtaQ- 
20 dkkrjla ta Sca^trj^ata^ aW sCg ikattov xal avi607tXatfi, 
td ccQa TtaQakkriXa t&v Sia^ti^fidtcov i^ ajto^f^^atog 
bQCD^sva avL6onkatfi (paCvstai, 

otiro fiaV, si iv rra ait^ ijtcTtiSG} ro fi^[ia rc3 
bQcofiivG) xioLto^ SL Ss liStscoQ^tSQOv strj ro oftfic^, ovtcog, 

25 ^6tc3 ydQ tb K^ Kal ^x^^ ^^^ ''^^^ ^ ^^^ ''^^ iTto- 
xsC^svov iitCitsSov xdd^stog fi KA^ ajtb Ss tov A iTtl 
ti\v ZA 1] AM xal ix^s^kiq^d^G) ijtl tb O, xal JtQ06- 

10. ^Z] Z corr. in E m. rec. V. 11. Ante 6'ftfta add. 

TCi ds nccQCcXXriXa xa ISJA, UN, BJ V. 12. ^yysiov V. 14. 

KS] !S} corr. in Z m. rec. V; item lin. 15, 16, 17. 16. net- 

^op Y. ISIKA'] ^A V. yavioc] in ras. v. 'bno (alt.)] 



OPTICORUM RECENSIO THEONIS. 



161 



sit enim F^d = KAj oculus autem sit J5, a quo 
radii adcidant B^, BA, BK^BF. itaque F^ ab angulo 
maiore cemitur quam KA, ergo F^ maior adparet 
quam KA [def. 4]. 

6. 
Longitudines parallelae, quae e distantia cemuntur, 
latitudinem inaequalem habere uidentur. 

sint enim longitudines parallelae BFy AZ^ oculus 
autem sit K dico, BT^ AZ latitudinem inaequalem 
habere uideri, et latitudinem propiorem semper maio- 
rem adparere longinquiore. 

adcidant radii K^^ KA, KH, KN, KB, K^, et 
ducantur rectae SA, JJN, BJ. iam quoniam est 

B^ , J L SKA > nKN, 

etiam recta SA maior adparet quam 

IIN eadem de causa etiam recta 

UN maior adparet recta J5z/. itaque 

longitudines non iam parallelae uide- 

buntur, sed latitudinem diminuentes 

inaequalemque habentes. ergo longi- 

tudines parallelae, quae e distantia 

cemuntur, latitudinem inaequalem 

habere uidentur. 

ita igitur, si oculus in eodem plano positus est, 

quo id quod cemitur; sin oculus eleuatior est, hoc 

modo. ^ 

sit enim ^, et a JSl ad planum subiacens perpen- 
dicularis ducatur KA, ab A autem ad Z-^ recta AM 
et producatur ad O, radii autem adcidant KB, KHy 




om. V. 17. iist^ov V. 18. iist^ov v. 

27. Post ZA add. ndd-srog m. 2 v. 

Eaclides, edd. Heibeig et MengQ. 'Vll. 



22. (fccivovxcci v. 



W 



162 OPTICOBUM BECENSIO THEONIS. 

ninrit(o6av &kxIvbq at KB^ KH^ KZ^ KJ^ KN^ KAy 
xal insie^dx^co^av at KM^ KS^ KO. ixel ohv djtb 
(letecoQOtiQOv tov , K inl tb M ixi^evxtav fj KM^ xd- 
d^atog aga i6tlv ijtl ti^v MA. 6iio£(og tfi) xal ii KS 

5 inl tiiv HN^ ^ 8\ KO inl ti^v BA. 6Q^oy6via aQa 
i6tl tdt KMA^ KSN^ KOA tQiycova. xai i6tiv ii 
lihv SN tfi MA l6ri' 7taQaXXriX6yQaii(iov y&Q tb MN' 
ixatiQa 8h t&v SKy KN iisi^aiv i6tlv ixatiQag t&v 
MK^ KA. iiei^Giv &Qa xal yoovla ^ 'bnb MKA tfig 

10 imb SKN. iieliov &Qa 6g)dij6etaL xal tb MA tov 
SN' biLoCiog xal tb ZM tov HS. &6te xal 8Aiy i^ 

\i ZA oXi]g tijg HN (lei^aiv fpaCvetai, 8i& t& aitA 8if 
xal ij HN trjg BA. &vi607cXatfi aQa xal oi)t(o (paC- 
vetai t& (leyi^rj. 

16 ^L r. 

T& iitl tijg ainijg eid^eCag '6vta t6a [leyid^ri 7tOQQ(o- 
riQ(o &kkiikc3v te^ivta &vi6a (paCvetat. 

i6tcj y&Q t6a iieyid^rj t& BF^ z/Z, Sftfte^ 8h i6t(o 
tb K^ xal &jtb tov biiiiatog tov K jtQo67tvjttitco6av 
20 6rljevg at KB, KF, KA, KZ* dQd^ii Sh i6t(o ij inb 
KZB ycovCa. ovxovv iieC^wv i6tlv ij £ ycjvCa tfjg O. 
&6te xal ij AZ iieC^wv (pav/i^etav tfjg FB. &vi6a 
&Qa (paCvetai t& BF^ AZ iieyidi]. 

2. KlSl] corr. ex KZ m. rec. V. 3. Ante Ttdd-srog add. 

i} KM m. rec. V, idem post iariv (lin. 4) m. 2 v. 4. MA] 
supra sc^^ Z m. 2 v. 6. iatl] iativ Vv. 8. iisi^ov v. 9. 
pistSov V, corr. m. 2. 10. fisitov — 11. HlS!] om. Vv. 11. 
ZM] ISIM p. HlSi] U^ p. 13. %al ovrco] om. Vv. 14. tcc 
{Lsys^ri] om. V; xai ovt€o tcc iisys^ add. m. rec. 17. Supra 
&XXi^Xa)v add. firi i(ps^fjg &XXi/jXoig m. 2 v. Post tsd^ivtcc add. 
Ttcel &vLaov Sis6t7i%6tcc tov 6'ft^aroff m. 2 v. 21. fisi^ov v. 

22. ^LSiiov V. 23. Post iisysd^ri add. tcc &qcc tccc asys^ri tcc 
inl ti\g aijtfjg svd^sLccg 6vtcc TCOQQmtSQOv ccXXi^Xtov tsd^svta &vlccc 
fpalvstaL m. rec. V. 



OPTICORUM RECENSIO THEONIS. 



163 



KZ, KJ, KNy KA, et ducantur KM, KS, KO. iam 
quoniam a puncto K eleuatiore ad M ducta est KM, 

ad MA perpendicularis est. 
eodem modo etiam KS ad 
HNy KO autem ad B^ per- 
pendicularis est. itaque tri- 
anguli KMAy KSN, KOA 
rectanguU sunt. est autem 
SN= MA (nam MN parallelo- 
grammum est). et SK > MK, 
KN>KA. itaque 

L MKA > SKN. 
quare etiam MA maior ad- 
parebit quam SN [def. 4]. 
similiter etiam ZM maior quam HS» quare tota ZA 
maior adparet tota HN. eadem de causa etiam HN 
maior quam BA. ergo sic quoque magnitudines lati- 
tudinem inaequalem habere uidentur. 




7. 
Aequales magnitudines in eadem recta positae^ si 
inaequaliter distant^ inaequales adparent. 

sint enim magnitudines 
aequales BF^ ^dZ, oculus 
autem sit K, et ab oculo K 
adcidant radii KB, KF^ 
K^^ KZ] angulus autem 
KZB rectus sit. itaque erit 
L2J>0. 

quare etiam AZ maior adparebit quam FB. ergo 
magnitudines BFj AZ inaequales adparent. 




164 OPTICORUM RECENSIO THEONIS. 

Tcc l6a ^syid^ &vv6ov Su6trjx6ta ovx avaX6y(og 
totg &7C06tiliia6iv bQatav. 

l6t(o y&Q tb Br rc3 z/Z t6ov zal KBC^^fo ai)t^ 

5 TCaQakkrikov^ Siiiia Sl §6t(o tb K^ xal ait^ aitov %qo6- 

nvjctitfo^av StlfBcg at KZF^ KB^ K^^ S)v ij KF TiQbg 

dQd^&g tfi FB i6t(o. (prnil Sij^ Ztv oix avaX6y(og q>a- 

vij6etaL t& BF^ jdZ (isyid^rj totg FK^ KZ Sia^tiliia^LV. 

inal y&Q 6q%^ i6tvv fj ijtb ^ZK^ 6%Bta aQa i6tlv 

10 fi {)7cb Z®K' &6t€ xal ^ &K trjg KZ i6tc lui^^ov. 
6 aQa TcivtQO) tp K^ Sta^tijiiatt Sh tp ®K xihckog 
yQag)6^6vog {msQ7t£6ettai tijv KZ. yeyQccg^d^cj Tcal i6t(o 
6 E®H. xal iTcel tb ®jdK tQCycovov iiei^ova X6yov 
i%ev TCQbg tbv 0EK to^iia ^TCeQ tb Z®K tQCycavov 

15 TCQbg tbv H®K tofiia^ ivaXXa^ aQa tb ®^K tQCycovov 
TCQbg tb ZSK tQCycovov [leC^ova X6yov i%eL ^TCeQ 6 
E@K tofieifg JCQbg tbv H®K toiiia» 6vv^ivtL ccQa 
tb ZjdK tQCycDvov TCQbg tb Z®K tQCycovov ^eC^ova 
Xoyov i%ei VjneQ 6 EHK tofievg TCQbg tbv H@K to^ia. 

20 (JAA' G)g tb Z^K tQCymvov JCQbg tb Z0K tQCycovov^ 
ovtcag ri ^Z TCQbg Z0, d)g Se 6 HEK tofievg scQbg 
tbv H®K toiiia^ oiitc^g fj iyjcb AKZ ycovCa TCQog tijv 
^Tcb &KZ. iv [leC^ovL k6ycp aQa i6tl xal rj ^Z TCQbg 
tijv Z@ i]7CeQ ij ZJ^ P y(ovCa TCQbg ti^v P yovCav. hg 

26 tf^ il AZ iCQbg ti{v Z®, ovt(og i^ TK TCQog ti{v KZ' 
xal ij KF aQa TCQbg tijv KZ iv [leC^ovL I6y(p i6tlv 
^TCeQ ii ZJ^ P ycovCa TCQbg ti^v P ycovCav. xal ix lihv 
tfjg 27, P ycDvCag tb AZ bQatat^ ix Sh tr\g P ycovCag 

2. civLGov] xal &VL60V T; supra add. Ttccl TtaQoiXXTiXa m. 
rec. V, naQokXriXa m. 2 v. Supra 0^3% add. aitb t&v 6(itidi- 

xoav m. 2 v. 3. ScTtoati^iLaaLv] corr. in 8ia6tif^yMGiv m. rec. V. 



OPTICORUM RECENSIO THEONIS. 165 

8. 
Magnitudines aequales inaequaliter distantes secun- 

dum proportionem distantiarum non cemuntur. 

sit enim Br= jdZ^ et ponantur parallelae, oculus 

autem sit K, et ab eo radii adcidant KZF, KB, K^d, 

quorum KF ad FB perpendicularis sit. dico igitur, 

magnitudines BF, ^dZ secundum proportionem distan- 

tiarum FK, KZ non cemi. 

nam quoniam L ^ZK rectus est, i Z0K acu- 

tus est. quare etiam ®K > KZ, itaque circulus 

centro K^ radio autem 0K 
descriptus rectam KZ ex- 
cedet. describatur et sit 
E®H, et quoniam est 
@^K:@EK>Z®K:H@K, 
permutando erit 
®^K:Z@K>E®K:H®K 
componendo igitur 
Z^K:Z®K>EHK:H®K 

est autem Z^K: Z®K= JZ : Z®^ et 

HEK : H®K = L ^KZ : L ^KZ. 

itaque erit jdZ: Z®> H -^- P: P. uerum 

jdZ:Z® = FK : KZ. itaque etiam 

Kr:KZ>2:+ P:P 

et ex angulo 2J -f- P cemitur ^Z, ex P autem angulo 

6. KZr] corr. ex KZ m. rec. V. 8. dtaGTi^tiocaiv] om. v. 

10. Z0K] G corr. v. ictiv p. /ifitfov v. 11. 6^ postea 

ins. V. t^ (pr.)] corr. ex t6 m. rec. V. 13. iTtst] iTci v. 

14. T(5v] corr. ex ti}i; m. rec. Vv. 15. ivctkd^ V. 21. 

^Qog (pr.)] om. V, nqbg tr^v m. rec. 22. ^n6'\ m. rec. V. 

^JCZ] JKHY. 23. iativY\. 28. P (alt.^li^Qst t^^.V 
Htt. V. 




166 



OPTICORUM RECENSIO THEONIS. 



ro BF. o{>x avdXoyov &Qa tots icno6rriiiLa6v ta t6a 
^sysd^rj bQatai, 

T& 6Q^oy6via iisydd^rj i^ aTCoatTJ^atog 6Q6^sva 
5 7CSQcq>SQ7i tpaCvsxai. 

i6t(o y&Q bQ%oy(hvvov tb BF 
[ietisg ^stdc^QOv] ^| &jco6tYiiLatog 
bQAiLSvov, oixovv hcsl sxa6tov 
t&v 6q(o^sv(ov ix^v ti (irixog icTCo- 
10 6tijiiatog^ oi ysvd^isvov oixsti 
bQatai^ il ftiv F aQa ycovCa oix 
bQatav^ t& 6\ jd^ Z 6rjiista [idvov (paCvstav, b^oCcsg 
xal S(p' sxa6trig t&v Xovtc&v y(ovv&v tovto 6vii^6stav. 
&6ts 8Aov nsQV(psQ^g (pavifj^stav. 




15 



V . 



Tg)v xcctG) tov ^iiiiatog iTCVTCsdcov xsviisvcdv ta tcoqqo) 
listscoQdtSQa (pavsltav, 

^6tG) y&Q Sfifia tb B avca tov FK iicvjciSov xsC- 
[jvsvov^ a(p^ oh Hfiiiatog %Qo67Cv%titG)6av axtlvsg at 

20 J5r, BJ^ BZ^ BK^ S)v fj BK xA^stog i6tco ijcl ro 
iTCoxsCfisvov ijcCjcsSov, Xiyca^ otv tb fz/ rov z/Z 
^stscoQdtSQOV (paCvstav^ ro 8s z/Z tov ZK slXrl^pd^co 
\_yd:Q] ijcl tilg ZK tv^bv 6rjfistov tb E^ xal i^x^^ ^Qog 
dQd^S^g rj EH. zal ijcsl aC '6^svg jCQdtsQov JCQog tijv 

25 HE jCQo6jcCjctov6vv ijjcsQ JCQog tijv EF^ jCQo6jcvjctitco 
tri HE fi [ihv BF xatc: ro H ^rj^svov^ fj Sh B^ xatdi 

7. kexoig {LBtioiqov] m. rec. V. 10. ysvoiisvov Vp. 15. 

i'] V, la' mut. in tj5' m. rec. 16. ininsdov'^ Ttsi^isvcov^ V 

(iy, /9, co m. rec), nsifisviav inmsdcav vp. 17. q>ccvsltai\ 

gpa/rsrai vpy m. rec. V. 20. B/f^ A m x^ba. m. ^ y. BJf (pr.) 



OPTICORUM RECENSIO THEONIS. 167 

BF. ergo magnitudines aequales secundum proportio- 
nem distantiarum non cernuntur. 

9. 

Magnitudines rectangulae, quae e distantia cemun- 
tur, rotundae adparent. 

sit enim BF rectangulum et e distantia cernatur. 
itaque cum omnia, quae cernuntur, longitudinem quan- 
dam distantiae habeant^ ubi cum posita sint^ non iam 
cernantur, angulus F non cemitur, puncta autem ^, Z 
sola adparent. et eodem modo etiam in ceteris angulis 
hoc eueniet. ergo tota niagnitudo rotunda adparebit. 

10. 
Planorum infra oculum positorum partes longin- 
quiores sublimiores adparebunt. 

„ sit enim B oculus supra 

planum FK positus, a quo 
^ radii adcidant BF, B^, BZ^ 

BK, quorum BK B,d planum 
subiacens perpendicularis sit. 
dico, Fjd sublimius adparere 
quam z/Z et ^dZ quam ZK 
sumatur in ZK punctum 
aliquod E, et perpendicularis 
ducatur EH. et quoniam 
radii ad HE prius adcidunt quam ad EF^ ad HE 
adcidat -Bf in puncto H, J5z/ m A, BZ in. M, iam , 

— p. 168, 8. (paivBxai] in ras. m. rec. V (fuit . . . x o^xoDv i 

r&v Scitb tov . ofniatog Ttgbg tb . . iitlmSov iCQoeitiittova&v ). 

hv — p. 168, 6. JSTZ] om. v. 22. ZK^KZ in lac. 

m. 2 p. 23. ydo] om. p. 24. ^x% 'LY^ ^ N « 




168 OPTICORUM BECENSIO THEONIS. 

tb A^ ij 8\ B2, xccto: tb M, iitel ohv tb H tov A 
(istscoQdtsQOv^ tb Si A tov M, aAA' iv S i6tv ro JEZ, 
iv tovto) tb f, iv ^ 8e tb A^ iv to^vtco tb A^ iv & 
8\ tb M^ iv toiko) tb Z, St& Sh twv BT^ BA ii AF 
6 tpaCvBtui^ 8i& Sh t&v BA^ BZ fi ZA^ Sidc S\ t&v BZ^ 
\ BK ij KZ^ oifxovv ii ^lv FA tflg ZA iietecoQotSQa 
tpaCvetai^ ij Sh ZA r^^ ZK' to: ydp {>7tb iietecoQOteQcov 
axtCvGDV bQwiieva iietecoQdteQa (paCvetac, 

ia\ 

10 Tcbv avco tov Hiifiatos iitmeScov xeLiievov tic TtdQQoi 
tajteLvdteQa cpaveltai. 

i6t(o yaQ Sfiiia tb B Tcdtco rot) AZ imniSov xeC- 
lievov^ a(p* oi 7tQ067tL7ttetG)6av dxttveg at BA^ BF^ 
BZ^ i)v i^ BZ xdd^etos i6tc3 iitl tb i)7toxeCiievov inC- 

15 TteSov, Xeyo)^ or^ ro FA tov FZ tajteiv6teQ0v cpaCvetai. 
Sid Sri tb JtQoexted^hv d^e^hQrjfia taTtevvotdtri tcbv &7tb 
tov B Hfifiatog TtQbg tb AZ ijtCTteSov 7tQo6JtiJttov6&v 
dxtCvcDv i6tlv fj J5z/, fj Se BF tfjg BZ taTtecvoteQa. 
dlkd Std ^ev t&v BA^ BF dxtCvwv tb AF cpaCvetat^ 

20 Sid Se tG)v BF^ BZ tb FZ. tb AF ccQa ta7tecv6- 
teQOv tov rZ bQcctac. 



1. insl ovv] bis p. 2, tcJ (pr.)] icri x6 V. 4. i^ z/T] 
m. 2 p. 6. Zz/I z/Z V. 7.^ ZK'] K in ras. m. 2 v. 8. 
Post (pccivBxm add. x&v &Qa tidxa} xov (corr. ex x&v) 6'ft^aro9 
TtSiiisvmv Ttccl xcc k^fjg V. Mg. m. 1 V: TT. i% Sri xovxov (pavs- 
q6v icxL (Srt add. m. rec.) xd iTclitsSa. i% xo% ^lsgov d^scogovnsva 
HolXa (paivsxai. xs^siarig yag xf}g 6ipsoig yiaxd iiscov xov ini- 
ns8ov iv x& iisxsmgoa cpavSQOv xb Xsy6iisvov ctQOGSTi^Xrid^ivxog 
xov TK iTtiTtsdov iiti xd ScQLaxsgd, mcxs ytal slg xd ds^id xa 
^6ppcii iiQoasxsiv nal sig xd dcQtaxsQd. sl yccQ fisxsoDQ^xsQa xa 
^xpa, SfjXov, 8x1 x6 iiiaov uoUov. 9. ta'] mut. in tp' m. 



OPTICORUM RECENSIO THEONIS. 



169 



quoniain H sublimius est quam A, A autem quam M, 
ubi autem H est, ibi est f, ubi A^ ibi A^ ubi M^ 
ibi Z, per J5r, J5z/ autem AT adparet, per J5z/, BZ 
uero Z^, et KZ per J5Z, BiC, sublimius adparet 
TA quam Zz/, ZA quam ZJST; nam quae a radiis 
sublimioribus cernuntur, sublimiora adparent [def. 5]. 

11. 

Planorum supra oculnm positorum partes longin- 

quiores demissiores adparebunt. 

oculus enim sit J3 infra pla- 
num AZ positus^ a quo adcidant 
radii J5z/, J5F, BZ^ quorum 
BZ ad planum suppositum per- 
pendicularis sit. dico, TA de- 
missius adparere quam TZ. 
propter theorema supra exposi- 
tum BA e radiis a B oculo 
ad planum AZ adcidentibus 

maxime demissus est^ BT autem demissior quam J3Z. 

uerum per radios BA,BT adparet AT^ TZ autem 

per BTy BZ. ergo AT demissius quam TZ cemitur. 




rec. V. 10. ini^sSov (corr. m. rec.) %SLftivG}v V, add. fi — a 
m. rec; %siii,iva>v imnidcov yp. 11. tpavsttoci] tpalvetat p et 
m. rec. V. 13. BJ, BF] BT, BJ V. 14. &v — 16. ^6<fe- 
Qriiia] m. 2 p, om. v, &V — 21. dQatai] in ras. m. rec. V; . ^j. 
a m. 1 fiiit: o^btioijv taitsivotdtri t&v &'7th tov B byi^tog ytffbg \t 
t6 JZ ininsdov ^qoGiivjttovG&v &7itivoDv iatlv ii BJ^ xal &7C^8- ^ 
Qov (paivstai tb J' tb J &Qa taitsiv&tSQOv (paivetai to4) JT, «& 

dh r tov z. 16. rj] z/r V. 20. t6 rz] om. v. jr} 

P m. 2 p. 21. Post dQ&tai add. t&v &Qa avm to% tffifumff 
%sipiiv(Dv xttl ta k^fjg Y, tb J &Qa tansiv&tSQOv tpalifetai t99 Fg 

t6 ds r toe Z mg. m. 2 p. ial 



170 



OPTICOEiJM BECENSIO TflEONIS. 



T&v slg toviiTtQO^d^sv (ifjxog ix6vt(ov toc [ihv iv totg 
Ss^iotg slg td^ &Qv6tBQcc doxst TCUQY^x^^av^ tSt dh iv totg 
&Qi6tSQotg slg t& Sa^id. 
6 i6t(o y&Q 6Q6[i€va t& BF^ jdZ^ 
Siiiicc Si tb JK, &q)' oH nQ06%L7Cti- 
tc36av "di^evg aC KF^ KA^ KB^ KZ^ 
KH^ Kjd. oixovv ro jd TcaQiix^av 

80X€t €lg t& &QL6teQ& iJTtBQ tb H. 

10 b^OLfog 81 xal tb B elg t& Se^v& 
Soxet naQr^x^ai ^iieQ tb A. &6te 
t&v elg toviijtQO^d^ev iirjxog ix6v- 
tc3V t& iihv iv totg Se^votg eig t& &Qi6teQ& Soxet 
TtaQfjx^av^ t& Sh iv totg &Qv6teQ0tg elg t& Se^vd, 




15 



cy 



T&v t6G}v [leyed^&v iTtb tb '6^iia xsv^ivG)v ta 
7t6QQG) xBLiisva iiEtsc3Q6tSQa cpaCvstav, 

i6ta} y&Q i'6a [isyid^ t& BF^ jdZ^ KA 'vnb tb 
ofifia tb N xsL^sva^ xal &%b tov N &iiiiatog 7tQ06- 
20 7tL3ttitci}6av &xttvsg aC BN^ NA^ NK oixovv [istsmQO- 
tdtrj i6tlv fj NB tcbv XoLTtcbv &xt£vcov' &6ts xal tb B 
6rj^SL0v. tb ccQa BF tov AZ iistsc3Q6tSQOv cpaLvstaL, 
tb Ss AZ tov KA, t&v aQa t^cov fisysd^&v 'bitb tb 
bfifia xsLiiiva)v ta 7t6QQG) xsLiisva [istscoQ^tSQa q^aCvstaL, 

25 lS'. 

Tcbv l'6a)v [isysd^&v &vc3 rot) bfi^atog xsLiiivc3v t& 
7t6QQC0 xsLfisva ta7tSLv6tSQa cpaLvstaL. 

3. Si] S' p. 7. al\ Xsya) oxi al v. 8. KK\ KN V. 

9. H] N V. 12. Toij^TtQoad^s V. ixtiivtav v, sed corr. 
IS, Seiiotg — 14. rofff] om. v. 18. KA\ om. v. 



OPTICORUM RECENSIO THEONIS. 171 

12. 

Magnitudinuni; quae ad partes anteriores uersus 
longitudinem habent^ ea^ quae ad dextram posita sunt, 
ad partem sinistram cedere uidentur, quae autem ad 
sinistram posita sunt, ad partem de^rL 

cemantur enim BF, jdZ, oculus autem sit Kj a 
quo radii adcidant KF, KA, KB, KZ, KH, K^. ita- 
que punctum J ad partes sinistras cessisse uidetur 
magis quam H, similiter autem etiam B ad partes 
dextras cessisse uidetur magis quam A, ergo magni- 
tudinum, quae ad partes anteriores uersus longitu- 
dinem habent^ ea^ quae ad dextram posita sunt^ ad 
partem sinistram cedere uidentur, quae autem ad 
sinistram posita sunt^ ad partem dextram. 

13. 
Magnitudinum aequalium sub oculo positarum 
longinquiores sublimiores adparent. 

sint enim BF, jdZ, KA 

magnitudines aequales sub oculo 

JV positae, et ab N oculo radii 

adcidant BN, NA, NK. ita- 

que NB reliquis radiis sub- 

limior est; quare etiam punc- 

tum B. itaque BF sublimior 

^ adparet quam AZ, AZ autem 

quam KA. ergo magnitudinum aequalium sub oculo 

positarum longinquiores sublimiores adparent. 

14. 
Magnitudinum aequalium supra oculum positaram. 
longinquiores demissiores adpareiLt. r^ 




172 



OPnCORUM RECENSIO THEONIS. 



s6t(o t6a ^syid^rj t& KN^ AZ^ Tjd &vcj tov 8/it- 
(latog xei^avu tov J3, tucI 
&7C0 tov B a^iiatog tcqo^- 
7tL7Cth(o6av &xttves cct J5JV, 
5 J3Z, 5z/. oixovv taTCeivo- 
tdtrj i6tlv fi J5z/' m6te xal 
tb jd. &6te xal tb [ilv Iz/ 
taicetvdteQov q^aivetai tov 
AZ^ tb 8\ AZ tov KN. 




10 



f 

le . 



"06a aXkrjXfxiv i)jceQe%eL t&v {>7cb tb Hfiiia xeiiievcov^ 
7CQ06i6vtog ^hv roi) ^ii^atog iieL^ovv tb iTCeQq^aLvdfie' 
vov (paLvetaL iieT^ov^ &jCL6vtog 8\ iXattovL fist^ov, 
i6ta) yaQ fiet^ov tb BF tov ®Z, xal (ifi^a xei^d^ca 

16 tb K &VG) t&v J5r, @Z, xal 7CQo6mjctetco &xtlg Sl& 
rov ® ij Kjd, ovxovv tb BF roi) @Z [let^ov q^aCvetaL 
ta B^' l'6ov y&Q ifpaCveto ro ®Z ta z/r*, iTCeLSij i)7cb 
tov avtov 6^^atog xal trjg K^d &xttvog ec^Qccto. ^&Xlv 
Sij iLetaxeC^d^m tb bfiiia inl tb A^ xal Sl& tov S 

20 jCQo6jCL7Ctstco &xtlg fj AN. ovxovv TcdkLV tb BF tov 
SZ ^et^ov fpaCvetaL rw BN. ildttovL aQa ^paCvetaL 
'bneQSxov tb BF tov @Z &7CL6vtog tov (iii^atog ^eQ 

7CQ06L6vtOg. 

Lg\ 
25 "06a &XXi^kov insQsiSL xdtc^ roi) b^iiatog xsl^svov^ 
7CQo6L6vtog filv tov ^iifiatog ikdttovL ^st^ov tb vtcsq- 
g)aLv6fisvov fpaCvstaL^ &7CL6vtog S\ [isC^ovl ^st^ov, 

3. B] m. rec. V. 7. J — rd] om. pv. &ats Y.cci\ 

m. 1 V, xai Sicc tovro m. rec. 9. Post KN add. tcbv ocqu 

tccDV fisysd^&v Hal tcc k^fjg m. rec. V. 13. icjti^vtog] -ov- in 

ras. V. 17. tdi (pr.)] t6 v. tcQv\ m. rec. V, comp. m. 1. 



OPTICORUM RECEN8IO THEONIS. 



173 



sint KN^ AZy Iz/ magnitudines aequales supra 
oculum B positae^ et a J3 oculo radii adcidant BN, 
BZj B^. itaque J5z/ maxime est demissus; quare 
etiam jd. ergo etiam Iz/ demissior adparet quam AZ, 
AZ autem quam KN. 

15. 
Magnitudinum sub oculo positarum, quae inter se 

excedunt, excedens oculo adpropinquante magis ex- 

cedere uidetur, recedente uero minus. 

sit enim J5r> 0Z, et oculus K supra BTy ®Z 

ponatur, radius autem KA per @ adcidat. itaque BF 

magnitudinem @Z mag- 
nitudine J5 ^ excedere 
uidetur; adparebat enim 
®Z = z/r, quoniam ab 
eodem oculo radioque 
KA cernebatur. rursus 
igitur oculus ad A trans- 
feratur, et per adcidat 

radius AN. itaque rursns BF magnitudinem &Z 

magnitudine BN excedere uidetur. ergo BF oculo 

recedente minus excedere uidetur magnitudinem &Z 

quam oculo adpropinquante. ^) 

16. 

Magnitudinum, quae oculo infra posito inter se 

excedunt, excedens oculo adpropinquante minus ex- 

cedere uidetur, recedente uero magis. 

1) Praeter nostram figuram, in qua m. rec. adscripsit ro0ro 
dgd-dv^ aliam quoque dissimilem habet Y. 

Ante QZ ras. 1 litt. v. 19. r<5 (pr.)] tr6 K v. 21. Ante 
S Z ras. 1 litt. v. iXdtttovt v, et V, aed cott. "ox. TSft, - ^j 




174 



OPTICORUM RECENSIO THEONIS. 



^B 



l6t(o iiettov tb BZ tov 0K^ xal tov A Hii^atog 
Tidtfo xsLiidvov 7CQ06M7Ctit(D Axtls ii AF 8l& tov ®. 
oixovv tb BZ tov 0K [ist^ov q)aCvBtav ta BF, [ista- 
7uC6^(o Sii tb A Siiiia ijtl ib JV, xal 7CQ067CL7ttit(o 
6 axtlg fi N^ 81&. tov G. cfocow tc&Xvv tb BZ tov ®K 
lietiov (paCvBtai rc3 B^, 7CQ06i6vtog [ihv &Qa tov H^- 
[latog iXdttovL ^Bt^ov (paCvBtaL imBQixov tb BZ tov 
SK^ aTCL&utog Sh (ibC^ovl. 

10 T9^a dXXi^Xfov vTtSQixBL tov ^iiiiatog iTC^ B^bd^sCag 
t& iXd66ovL [iByid^BL Svtog^ 7CQ06L6vtog ts Tcal d(pL6ta- 
[livov tov 6(iiiatog rp t6G) alsl d6^BL tb i)7CBQ(paLv6- 

(IBVOV tOV iXd660V0g VTCBQixBLV. 

i)7CBQBxit(o yaQ tb Bjd tov 
15 ®H tp J5r, xal i^CL^Bvxd^st^a 

4j r® ixfiBfiXifi6%^(o^ xal i6t(o 

tb ^iiiia iTcl tov Z. ovxoih/ i^ 

aTcb tov Z dxtlg 7CQo67cC7Ctov6a 

xatd tijv Z r kvsx^^^^BtaL. 
20 7cdXLV Sii ^BtaxBC6^(o tb ^fifia 

i7cl tov K ox^xovv S^d td avtd fj d7cb tov K Hiiiiatog 

dxtlg 7CQo67tC7Ctov6a xatd tijv KF ivsx^^^^staL. tc3 ait^ 

aQa v7CBQi^BL tb B^ rov &Hxal 7CQo6L6vtog tov ^iifiatog 

xal dq)L6ta^ivov. 
25 Lrj', 

Tb Sod^lv vil^og yv&vaL^ 7t66ov i6tCv. 

s6t(o ydQ^ Sst i7tLyvCbvaL vil^og^ 7t66ov i6tC^ tb 

BF^ xal 7tQ067tL7ttit(o dxtlg ijXCov Sid tov B ij B^. 

4. 8ri\ 8i V. TcgoGTanxho}] -an- in ras. V. 7. ^cettov v. 
11. ^sysd^ri V. Bvttog v, sed corr. 15. QH'] 0H Vv, 

9N p. 16. r@] r in ras. m. 1 V. 23. 0H] 0H Vv. 



® 



^d Ih 



K 

-\ — 



Z 



\ 



OPHCORUM RECENSIO THEONIS. 



175 




sit jBZ > &Kj et oculo A infra posito radius AF 
per adcidat. itaque BZ magnitudineixL ®K magni- 

tudine BF excedere 
uidetur. iam oculus 
^ ad JV transferatur, 
et per adcidat ra- 
dius N^d. rursus igi- 
tur BZ magnitudi- 
nem ® K magnitu- 
dine B J excedere 
uidetur. ergo oculo adpropinquante magnitudo ex- 
cedens BZ minus excedere uidetur magnitudinem ^K^ 
recedente uero magis. 

17. 

Magnitudinum, quae oculo in eadem recta posito, 
in qua est magnitudo minor, inter se excedunt, ex- 
cedens, siue adpropinquat siue recedit oculus, semper 
eodem spatio minorem excedere uidebitur. 

excedat enim Bjd magnitudinem 0H magnitudine 
BTj et ducta T® producatur, oculus autem in Z 
positus sit. itaque radius a Z adcidens per ZT feretur. 
iam rursus oculus ad K transferatur. itaque eadem 
de causa radius a K oculo adcidens per KT feretur. 
ergo BA eodem spatio magnitudinem %H excedet, 
siue adpropinquat siue recedit oculus. 

18. 

Datam altitudinem cognoscere, quanta sit. 
sit enim BT altitudo, quae quanta sit, cognoscere 
oporteat, et per B adcidat radius solis BA. rd ^s^ 




176 OPTICORUM RECENSIO THEONIS. 

oifxovv 6xlA ^6tai ii F^. iXaPov Snl rv yva)Qi[iov 
(laysd^og rb KZ xal ivilQiio6a imb rijv ^ ycoviav 
7CaQdXXi]Xov rfj B 17. oixoiyi/ 
i6riv^ &g rb ^r TCQbg ro JT-B, 
5 ovrc3g rb ^Z TCQbg ro ZK. xal 
yvmQLfiog 6 l6yog 6 rfig ^Z TCQbg 
ZK' yvfDQi^og aQa xal 6 rrjg 
^r JCQbg FB. %al idri yvtoQt- 
^og fi ^ r 6xt(i' yv^QVfiov aQa xal ro FB vtl;og. 

10 Ld'\ 

Mii ivrog iiXCov rb dod^^v vilfog yv&vav^ iikCxov 
i6rCv. 

i6r(*} y&Q^ Sst ijciyv&vai v^og^ Jcr^XCxov i^rCv^ 
ro Br^ xal xsC^d^o) xdrojcrQOv ro KA^ ififia Sh ierto 

15 ro jd^ xai dx^ airov jCQ06m%rir(o dxrlg ii jd& xal 
dvaxsxkd^d^cj &g rj ®B ixl ro B iciQag^ xal d^cb rov /i 
'oy^yiarog xd^srog ri ^Z. oixovv t6ai sl6lv al TCQog 
rc3 S ya}vCaL dkkTf^kaig' rovro ydQ SsCxvvrav iv rolg 
KarojcrQizotg. dXXd xal fj jCQbg ra F ry TCQbg rc3 Z 

20 t6r} ierCv dQd^i^ yaQ i6riv ixariQa air^v. KotTCr^ aQa 
rj TCQog rc5 B Koinfi r§ %Qbg rc3 ^d t6ri i6rCv. &6rs 
SfLOLov av sl'rj ro BF® rQCyovov rp ^dZ® rQLythvp. 
s6rLv aQa^ cag ^ ©F TCQbg FB^ ovtcsg ij &Z XQbg Z/l. 
rvig Ss ®Z %Qbg Z^ X6yog Sod^sCg i6tiv' xal tfjg SF 

25 ccQa TCQog FB yvtDQi^og 6 k6yog i6tCv. yv(DQi^og Sh 
^l ©r* yv^QL^ov aQa xal ro FB vipog. 



2. iviJQiioaTocL V. Ante J add. ngbg ra m. rec. V. 4. 

FB] Br p. 8. ictLv Vv. 9. amd' yvdoQiiiov] in ras. m. 1 V. 

Post v^ipog add. rb dQoc Sod^lv vipog ^yvtoarui ndaov iarlv 

m. rec. V. 13. iari p. 15. Supra dG add. rc5 %ur6nrQ(Q 



OPTICORUM RECENSIO THEONIS. 



177 



tur umbra erit. sumpsi igitur magnitudinem aliquam 
notam KZ et eam in angulum ^ magnitudini BF 
parallelam aptaui. itaque est ^F: FB — jdZ: ZK, 
et ratio jdZ : ZK nota est; quare etiam ratio jdF : FB 
nota. et umbra jdF nota est. ergo etiam altitudo 
FB nota. 

19. 

Sine solis usu datam altitudinem cognoscere, 
quanta sit. 

sit enim BF altitudo, quae quanta sit, cognoscere 
oporteat, et speculum coUocetur KA^ oculus autem 

sit ^, et ab eo radius 
adcidat ^® et inflectatur 
ad terminum B ut ®B, 
et ab oculo /j perpendi- 
cularis sit jdZ. itaque 
anguli ad ® positi inter 
se aequales sunt; hoc enim 
in Catoptricis demonstra- 
tur. uerum etiam 
LT^Z', 
nam uterque rectus est. itaque qui relinquitur angulus 
ad B positus angulo ad jd posito aequalis est. quare 
BF® r^ jdZe, itaque ©F: rB = ®Z: Z/l. uerum 
ratio (s>Z:Zjd data est: quare etiam ratio ®T: FB 
nota est. et ®r nota est. ergo etiam altitudo FB 
nota est. 

m. rec. V. 16. a»g — nsQag] del. m. rec. V, supra scr. &X9*^ 
ov avii^aXst t& nsgatL tov B T ybsys&ovg r© B m. rec. 18. t^ 
t6 V. 24.'Aiite X6yog add. 6 m. rec. V. iati p. 25. 

iatL p. 




Euolides, edd. Heiberg et Menge. 'Vll. 



VL 



178 



OPTICORUM RECENSIO THEONIS. 



X 



Tb Sod^lv pdd^og ixiyv&vav^ 7Ci]lvxov itsxCv, 

l6ta) yoLQ ro fidd^og^ 8 8st ijtvyv&vat^ nrikCxov i6tCvy 

tb KB^ xal xsC^d^o) ii^[ia tb ^, xal nQ067Ci7Ctit(o axtlg 
6 i^ ^AK slg tb pdd^og^ xal ^^-O-o 

axb tov ^ TCaQ^ tijv BK f} ^Z. 

iTcel jcaQakkriX6g i6tiv ii BK tfi /JZ^ 

Tud iiLTcintioxsv ii jdK^ t&g ivaXld^^ 

ymvCag tccg i)7cb BKA^ AjdZ t6ag 
10 iXl^^XaLg noLsl, sl6l 8h xal av xat& 

xoQvtpiiv at JCQbg tp A l6ai (iAAiJ- 

kaig^ xal ii Xoijcii &Qa yc3vCa r^ 

Aotjr^ t6ri ietCv. l6oyAviOv aQa i6tl 

tb BKA tQCy(ovov t& A/JZ tQi- 
16 yd>V(p, i6tiv &Qa^ cag ij AZ iCQog 

Z/l^ i\ AB TCQbg BK. Sod^elg Si 6 tflg AZ jCQbg Z^d 

X6yog" Sod^elg aQa xal 6 tfig AB TCQbg BK X6yog. xaC 

i6tv So%^et6a fi AB' Sod^et^a aQa xal fj BK. 



B 


A 


A 




/ 




K 


r 



xa . 
20 Tb So%t\v iiTJxog iicvyvcbvav^ TcrjXCxov i6tCv. 

e6tc3 ydQ^ Set iifjxog imyvcbvav^ TcrjXCxov i6tCvj 
tb BF. xeC^d^cj di) H^^a ro ^, dq)^ ov XQ067Cvntito- 
6av dxttveg at jdB^ /iT^ xal djcb rov Z ^x^^ TCaQa 
tijv BF ii ZK. oixovv i6tvv^ hg ii ZK iCQbg K^j 
26 ij BF ^Qbg Fjd. yvmQVfvog Sh 6 tijg ZK TtQbg iCz/ 
k6yog' yv(hQv^og aQa xal 6 tfjg BF iCQbg TA },6yog. 
xal yvd)QV^og ij fz/* yvd^QV^og ccQa xal rj FB. 

3. iGtlv] iati Vp. 4. KB] coir. ex KF v. nQoa- 

TtLTtzho V. 5. t6 ^dd^og] mut. in tb TtsQag tov ^dd^ovg m. 

rec. V. 6. Supra nciQd add. ijtoL TtccQciXXriXog m. rec. V. 



OPTICORUM RECENSIO THEONIS. 179 

20. 

Datam profdnditatem cognoscere^ quanta sit. 

sit enim KB profanditas^ qnae quanta sit^ co- 
gnoscere oporteat, et oculus ponatur ^, radius autem 
^AK ad profunditatem adcidat, et a ^ rectae BK 
parallela ducatur ^Z. iam quoniam BK rectae ^Z 
parallela est, et ^K in eas incidit,. angulos altemos 
BKA, A^Z inter se aequales efficit. uerum etiam 
anguli ad A uerticem positi inter se aequales sunt; 
quare etiam reliquus angulus reliquo aequaKs est. 
itaque trianguli BKAy A^Z aequianguli sunt. quare 
est AZ : ZA = AB : BK uerum ratio AZ : ZA 
data est; quare etiam ratio AB : BK data. et ^^ 
data est. ergo etiam BK data est. 

21. 
Datam longitudinem cognoscere, quanta sit. 

sit enim BF longitudo, quae 
quanta sit,. cognoscere oporteat. 
iam ^ oculus ponatur, a quo 
radii adcidant ^5, ^r, et a Z 
ducatur ZK rectae BF par- 
aUela. est igitur 

ZK:KA^Br:r^, 
^^ uerum ratio ZK: K^d nota est; 
itaque etiam ratio BF^r^ nota est. et FA nota 
est. ergo etiam FB nota est. 

7. Post iitBi add. ovv m. rec. V. 9. BKA\ B e corr. m. 1 V. 

10. df ] corr. in di} p, di} Vp. 13. ioxL^ iatlv v. U. t6] 
rm V. B KA] corr. ex B KA m. rec. V. 21. ft^xogl in ras. m. 1 V. 

' iGxiv^ comp. V, iGxi p. 22. /f] e corr. m. 1 V. 28. Siqm, 
TiaQa. add. TtoiQaXXTikog m. rec. V. 24. ZiiC(alt.W K.\&.tm^'^| 





180 OPTICORUM RECENSIO THEONIS. 

^Eav iv r^ avr& ijttTtsdp^ iv a ro S^^a^ xvockov 
%BQiq)iQeia r^-O-jJ, avQ^ala yQa^iiif ij rov xvxXov tcsql- 
(piQSLa (pavalrav. 
6 a6r(o yaQ nsQifpiQaia 'fl BF^ b^^a 8a rb ^ iv rc3 
air^ ijtLTtidG) 8v rfl BF icaQKpaQaCa^ «9' o5 jtQ06- 
^LTtriro^av iitl;aig at 
JB^ Z^, jr. o\m- 
ovv^ iital r&v bQCD^i- 
10 VC3V ovShv a^a bQarav^ 
oix av (paCvovro fi ZB 
TtaQLtpiQaia^ ric 8a Z^ B xiQara. do^av &Qa i^ ZB 
TtaQifpiQava aid^ata alvai, b^oCcjg dh xal ii ZF. okrj 
aQa ij BF jtaQtq^iQaia svd^ata Sd^ai slvai. 

16 xy\ 

2Jg)aCQag 67t(o6ovv dQcsiiivrjg vito rov avbg o/x/xo^ro^ 

aXarrov aCal ii^i6(paiQCov d^pd^TJ^arai^ airb Sh ro 6qg)- 

IJLSvov rfjg 6(paCQag vjtb xvxkov 7taQLax6^avov (paCvaraL. 

^6rcj yaQ 6(patQa^ ijg xivrQOv l6rcj ro K^ (i^^a Sh 

20 i6rc3 ro 5, xal ijta^avx^c3 ij BK^ xal TtQbg dQd^ag 
airfj ^x^^ ^*^ ''^^^ ^ V ^K/d^ xai ix^a^lri^d^ai ro 
Sl& r&v BK^ FKjd ijtCjtsSov noLiqesL Sij iv rij 6(paCQcc 
xvxXov. TtOLsCrG) Sii rbv FjdAN^ TtsQl Ss riiv KB [dta- 
^LsrQOv^ xvxkog ysyQciq^d^cj^ xal ijts^svx^c36av at KZ^ 

25 ZB^ BA^ AK^ AZ. ov;cow ijtsl dQd^aC sl6lv at vTtb 

4. qpavffrat] corr. ex (palvsraL m. 1 V. 6. ro] tc5 v. 6. 
ov] in ras. m. 1 V. 9. iTtsL] inl v, et V, sed corr. 12. rcc 
ds] mut. in ScXXu iiovoc rd m. rec. V. 17. &sl p. 19. ^ata 
(alt.)] del. m. rec. V. 21. ro] in ras. V. 22. FKJ] corr. 
ex i/ m. rec. V. 23. tcoisIxo v. rov] r6 v. FJAN] N 
mut. in Z m. rec. V, Z add. m. 2 p. dLaiisrQov] m. rec. V. 
^o. B^] corr. ex BJ V. 



OPTICORUM RECENSIO THEONIS. 



181 



22. 

Si in eodem plano, in quo est oculus, arcus cir- 
culi positus erit, arcus circuli recta linea esse ad- 
parebit. 

sit enim BF arcus, oculus autem jd in eodem 
plano positus, in quo est arcus BF^ et ab eo radii 
adcidant jdBj Z^, ^JT. quoniam igitur eorum, quae 
cernuntur, nihil simul totum cemitur [prop. I], arcus 
ZB non adparet^ termini autem Z^ B adparent. itaque 
arcus ZB recta esse uidebitur. et similiter arcus ZF. 
ergo totus arcus BF recta esse uidebitur. 



23. 
Quomodocunque sphaera ab uno oculo cemitur, 
semper minus hemisphaerio cemetur, ipsaque pars 

sphaerae, quae cernitur, cir- 
culo comprehensa adparet. 
sit enim sphaera, cuius 
centrum sit K^ oculus autem 
sit By et ducatur BKy et 
ad eam perpendicularis per 
K ducatur FK/Jy et pla- 
num per BK^ FK^ pro- 
ducatur; circulum igitur in 
sphaera efficiet. efficiat igi- 
tur r^AN, et circum 
^L^ circulus describatur, et 
ducantur KZ, ZB, BjI, 
AKy AZ. itaque quoidaia 
L KZBy BAK recti sui^t^ quia in semicirculis sunt^lH 
KZ, KA radii sunt, BA, £ Z in "vm.o ^^^ isSMiiiH 




182 OPTICORUM BECENSIO THE0NI8. 

KZB^ BAK 8i& rb iv '^ii.vxvxXioig elvai xal & xiv- 
TQOv rag KZ^ KA^ xaJd'* h/ 6riiutov iq^iipovtav aC 
BA^ BZ t^g 6fpaCQag' aC &Qa axb tov B 5|i&ftarog 
7eQ06xixrov6ac axrtvsg Tcarct rctg BZ^ BA 7Cs6ovvtau 
6 xal ixsl ixd6rri t&v XQbg rp S y(ovi&v dgd^ i6tv 
iut t6 xaQakkfikov slvac rij^v FA ry ZA^ xal tdri ^ 
Z@ rfl BA^ iicv Si^ ^oii^rig rHg 0B rb @ZB rQL- 
yeyvov %bqvbvb%%^v slg rb aifrb tc&Uv iatoTcarcuirccd^y 
od^BV fJQ^aro (piQBed^ai^ i^ rs BZ XBQupsQOiUvri xa%'* dv 

10 itpd^Brai, rf^g 6g)atQixijg i]Ciq>avBCag 7C€cr& t6 Z, Tcal 

xvxkog B6rav yByQafifiivpg Svo: r&v Z, A ^rnuCov. &6rs 

' imb xvxXov av jtBQiijOLro rb bQihiuvov rrlg 6q>aCQag^ 

S ys iXarriv i6riv '^^i6q)aLQCov' rb y&Q ZA iXarrdv 

i6rvv fifiLxvxXCov, &6rB xal rb imb rrjg ^soig tcbql- 

15 sx^iiBvov IXarrov i6rcv iific6g)aLQCov. 

xS\ 
Tov Sfifiarog 7tQo6i6vrog iyycov rf^g 6(paCQag ikar- 
rov s6rac rb bQo^^BVOv^ So^bl Sl ^st^ov bQa6^ac, 

B6r(o yccQ 6(paXQa^ ^g xivrQOV i6ra) rb JC, xal ccTtb 
20 rov z/ o^^ar og iscB^Bvxd^o) btcI rb xivrQOV ii ^K^ xal 
Slcc tov K TCQbg dQd^ctg i^x^ca ij BF^ tcbqI Sh rijv ^K 
xvxlog ysyQd^pd^cs^ xal ixB^Bvx^c36av at AN^ NK^ 
AA^ AK. ovxovv dQd^al seovraL at TCQbg rotg A^ N 
yc3vCaL Slo: t6 iv 'fj^LxvxlCG} BivaL* xa^' sv aQa i(p- 

5. S] e corr. m. 1 v. 8. slg r6] slg v. 9. qpi-] in 

ras. V. Post iv add. arnisZov p et m. rec. V. 13. o ys] 

mut. in TLCci m. rec. V. iarLv] mut. in i<nai m. rec. V. rd 
— 15. iiiiLG(paLQlov] mut. in ij yccQ ZA SLdpLSTQOs ovaa xov 
xi*xloi; rov $LaLQOvvtog rb dgmiLSvov rQg 6(paLQag iXdrrav iarl 
ri]g dr SLaiLStQOv o-ScTTjff rijg 6<paLQag m, rec. V. 13. ZA 

ZN Y , N supra scr. m. 2 p. 14. i6TL p. 7(sqlsx6(isvov 

oQmgisvov \ et snpra add. m. 1 p. IT. l^xi^vov V. 22. AN' 



OPTICORUM RECENSIO THEONIS. 



183 



contingent. quaxe radii a B oculo adcidentes per B Z^ BA 
cadent. et quoniam singuli anguli ad S positi recti sunt^ 
quia r^ rectae ZA parallela est, et Z0 = &Ay si 
manente ®B triangulus ®ZB circumuolutus ad idem 
punctum rursus restituitur, unde ferri coeptus est, 
BZ circumuoluta uno loco superficiem sphaerae con- 
tinget, scilicet in Z, et circulus per puncta Z, A de- 
scriptus erit. itaque quae cemitur pars sphaerae, circulo 
comprehensa eritj et minor est hemisphaerio; ZA enim 
minor est semicirculo. ergo etiam quod uisu com- 
prehenditur, minus est hemisphaerio. 



24 
Oculo ad sphaeram adpropinquante, quod cemitur, 
minus erit, uidebitur autem plus cemi. 

sit enim sphaera, cuius 
centmm sit iC, et a ^ 
oculo ad centmm ducatur 
AKj et per K perpendicu- 
laris ducatur BT, et cir- 
cum AK circulus descri- 
batur, ducanturque AN^ 
NKj AAy AK, itaque 
anguli ad A^ N positi recti 
erunt, quia in semicirculo 
sunt. in uno igitur puncto 
AA^ ^N sphaeram con- 
tingunt. itaque radii a ^ 
oculo adcidentes per dA^ 




corr. ex AN m. rec. V. 23. Post AK add. N/i m. 2 V^ 
m. rec. 




184 OPTICORUM RECENSIO THEONIS. 

dTCrovrai aC ^A^ ^N rfjg 6(pcciQag\, aC &Qa ajtb tov z/ 
^[i^atog TC(fo6%C%tovtSai dxttvsg xatii tctg ^A^ ^N 
7CB6ovvtai. nAXiv 8ii fistaxLveL^d^o tb ^ 8/it/ta iTcl 
tb P, xal tcsqI tijv PK xihckog ysyQccq^d^m^ otal iTts- 
6 ^stixd^oD^av at PZ^ ZK^ P2J, 2JK oimovv at PZ^ P2J 
xad'^ ?i/ ifpdjctovtav trjg 6(paCQag. Tcal aX ys &jcb tov P 
^[i^atog ixttvsg 7CQo67cC3Ctov6aL xata t&g PZ^ PU ns- 

1 6ovvtai. &6ts bQ&tai iycb fihv tflg P ytovCag tb ZS^ 
iTcb Sh tfig J tb NZA^fisttov Sh tb NZA tov ZS 

10 itstiv. q^aCvstav Sh ikattov psC^c3v yccQ i6tiv r^ P 
ycDvCa trjg ^ ycavCag^ tk S\ ^%b yisC%ovog ycnvCag 6q(o- 
[isva [isC^ova q>aCvstac, fist^ov aQa q^aCvstav tb Z2J 
tov NZA^ i6tv S\ iXattDv. 

M ■ A -' ■ 

xs\ 

15 ZfpaCQag Sia tcbv Stio d^^dtcov bQcafisvrig^ iav 'fj 
SvdfistQog tr^g 6cpaCQag t6ri y tfj sid^sCa t^ Sis6t(h6ri 
aicb tcbv d^fidtcavj ^^L6g)aCQtov avtr^g dcpd^r^^stac, 

s6tc3 y&Q 6tpatQa^ ^g Svd^stQog fi BF^ xal anb 
tG)v B^ r ^%%cQ6av TtQog hQ%&g «f BZ^ FA^ xal aicb 

20 tov Z f^i^^cj 7CaQ& tijv BF ii ZA^ xal xsC6d^c3 'sv ofi/ita 
STCl tOV Z, tb Ss StSQOV iTcl tov A^ &7cb Ss tov ^ 
xivtQOv V^x^^o TCaQa t^v BZ r^ ^K, oincovv i&v 
^svoii6rig tfjg jdK tb BK xaQakkriMyQa^^ov 7Csql- 
svsx^sv sig ro avtb Tcdlcv &7Coxata6ta%'^^ Sd^sv fJQ^ato 

25 q^SQS^d^ac^ tb 7CSQcyQa<plv ixb tf^g B^d ^xrjiia xvxkog 
i6tai^ 5g ys Sl& tov xsvtQOv i6tl trjg 6<paCQag. &6ts 

4. PK] P-p, KPy. 8. 6q-] in ras. m. 1 V. 9. rb 

NZA (pr.)] tbv ZA v; rb iVZ, add. 2A m. 2, p. rb NZA 

(alt.)] rbv ZA v; rb NZA, supra add. 2^ m. 2, p. 10. 

/arij/ (pr.)] iffti p. fisltov v. P] e corr. p. IS. NZA] 



OPTICORUM RECENSIO THEONIS. 



185 



^N cadent. rursus Igitur ^ oculus ad P trans- 
ponatur, et circum PJiL circulus describatur, ducan- 
turque PZ, ZK, PU, ZK. itaque PZ, PU in uno 
puncto sphaeram contingunt, et qui ab P oculo ad- 
cidunt radii, per PZ, PU cadeni quare ab angulo P 
cernitur Z27, a ^ autem NZA] est autem NZA>Z2J. 
uidetur autem minus esse; nam /. P> z/, et quae ab 
angulo maiore cemuntur, maiora adparent [def. 4]. 
ergo ZU maius uidetur esse quam NZA, est uero 
minus. 

25. 
Ubi sphaera ambobus ocuKs cemitur, si diametrus 
sphaerae rectae, quam oculi inter se distant, aequalis 

est, hemisphaerium cemetur. 

sit enim sphaera^ cuius 
diametrus sit BT, et a 5, T 
perpendiculares ducantur BZ, 
TAy et a Z rectae BT parallela 
ducatur ZA, et alter oculus 
in Zy alter in A coUocetur, a 
centro autem /1 rectae BZ 
parallela ducatur /IK itaque 
si manente /IK parallelo- 
grammum B K circumuolu- 
tum rursus ad eundem locum 
restituitur, unde ferri coeptum est, figura a 5^ de- 
scripta circulus erit, qui per centrum sphaerae pro- 

JNTZ, add. HA m. 2, p. lexiv Vv. 20. Supra ita^a scr. 

7\roi Tioi^dXXTiXQq m. rec. V. BT] post B ras. V. ZA\ 

corr. ex ZA m. rec. V. 21. z/ xfvr^ov] xcVr^ov A p. 22. Supra 
maqa, scr. •jia^aXX7\ko^ m. rec. V. idv\ in ras. m. 1 V. 26. 
off yf] 5 et y in ras. m. 1 V. «6x%"\ &- yq. ra*^. ^. V^ , 




186 OPTICORUM RECENSIO THE0NI8. 

rb fific6q>ai(fiov rijg 6<pa£(fag [i6vov dq^d^i^^sraL {>7cb r&v 
Z, A dmidrcov. 

^E&v rb r&v 6[iii,(irc3v Si&^rrnLa iLsltpv fi rfig Sia- 
5 fiirQOv rf^g 6<paiQag^ iiiiii6q>aiQCov [ist^ov rb 6(fd)[isvov 
rijg 6<paiQag 6(pdij6srat. 

i6r(o y&Q 6<palQa^ ^g xsinQOv rb JST, r&v Sh 6[i[id- 
rcav Svd^rriiia rb BF [ist^ov 6v rf^g Sta[iirQov rfjg 
6(paCQag^ xal Sl&^ rov K xal rrjg BF iK^s^M^^^^ca stcC- 

10 jcsSov Tcal jeocsCroi sv r^ 6(paCQcc xihcXov rbv ^ZN^ 
xal 7tQ067cmrirc36av dxrtvsg xa-O*' sv a7tr6[isvaL aC B^^ 
rZ. (ydxovv ix^aXX6[isvai 6v[i7Cs6ovvraL dXXtf^lacg^ 
ijcsiSii ii Br rfig iv rfj 6(paCQcc Sia^iirQov [isC^cav i6rC, 
6v[i7CL7crirc36av Sij xar(i ro @ 6ri[istov. oixovv i^csl 

16 aTcb rov @ 6rj[isCov aC ®Z^ Sjd xad'^ Isv i(pa7Cr6[isvav 
7CQ067CSJCr(x>xa6iv^ lka66ov av stri rb ZNjd ii[iLxvxkCov* 
at yaQ &ZK^ ®/JK ycovCac ^Qd^aC sl6vv. ro aQa Xol- 
7cbv rfig 6(paCQag [ist^ov fj[it6(paLQCov bQarac i)7cb rcbv 
B^, TZ. 

20 x^. 

'Edv rb r&v 6[i[idrc3v SLd6rri[ia iXa66ov jj rflg Sva- 
[lirQOv rrjg 6q>aCQag^ rb 6q(D[isvov rrjg 6(paCQag iXa66ov 
ij[ic6(pacQCov 6(pd"ifi6srai. 

i6rc3 ydQ ^cpatQa^ ^g xivrQOv ro iC, r&v Sh 6[i- 

25 [idrc3v Sid6rrj[ia rb BF ikarrov 6v rfjg Sca[iirQOv rrjg 

6(paCQag^ xal Sid rov K xal rfjg BF ix^sfiXil^d^co iTcC- 

TCsSov xal 7CoisCrco iv rfi 6(paCQcc xvxkov rbv ZHN. 



6. TjiviacpaiQLov v, et p, sed corr. 10. Ttoulto v. 11. 
dxrtpos Vj sed corr. iv] ov (yrjftaJov v, ariiistov add. m. rec. V. 



OPTICORUM RECENSIO THEONIS. 187 

ductas erit. ergo hemisphaeriuin tantum sphaerae ab 
oculis Z, A cemetur. 

26. 

Si distantia oculorum diametro sphaerae maior est, 

quae cemitur pars sphaerae, maior erit hemisphaerio. 

sit enim sphaera, cuius centmm sit Ky oculorum 

autem distantia sit BT maior diametro sphaerae, et 

per K ^i BT pla- 
num producatur et in 
sphaera circulum^ZiV 
efficiat , radiique ad- 
cidant 5^, TZ in 
uno puncto tangentes. 
productae igitur inter 
se concident, quoniam 
^rdiametro sphaerae 
maior est. concidant igitur in puncto 0. itaque quo- 
niam a puncto rectae 0Z, ®/i in uno puncto 
contingentes adciderunt, ZN^ semicirculo minor est; 
nam anguli ©ZK^ ®^K recti sunt. ergo pars reliqua 
sphaerae, quae a B^j FZ cemitur, hemisphaerio 
maior est. 

27. 

Si distantia oculorum diametro sphaerae minor est, 

quae cemitur pars sphaerae, minor est hemisphaerio. 

sit enim sphaera, cuius centrum sit K, distantia 

autem oculorum sit BF minor diametro sphaerae, et 

per JiL et -BJT planum producatur et in sphaera cir- 



17. Post yaQ add. 'bn6 m. rec. V. ^iai, p. 26. ^Xatrov\ 
-TT- in ras. m. 1 V. 27. tcoieito ^. xo-v^ ^^Tt. 'si.xVi^. 




188 



OPTICORUM RECENSIO THEONIS. 



ilxd^co^av Sl &7C0 t&v B^ F 6[i^(itc3v xad'^ ?v sg)am6' 

^evat al BZ^ FN %al 6viJLm7ttat(X)6av &XXT^kaLQ Tcatct 

tb &' 6viL7ce6ovvtai y&Q^ 

ijcscdrlTtaQ avL6oC ai^iv 
b T^ ta FB xal ij tfjg 

6q)atQag did^atQog. oifx- 

ovv al &7cb tov & 61]' 
^aCov 7CQo67cCxtov6aL 

TCQbg ti^v 6(palQav aXat- 
10 tov fi^L6q)aiQCov TCaQC- 

K7]ilfovtar tb aQa ZHN 
lXa66ov fiiii6(pacQCov 

i6tCv. &6ta tb vTcb tcbv 

Bj r d^^dtcov 6Q(h^a- 
15 vov lXa66ov av atri ij^c6q)acQCov. 




KvXCvSqov 67CCJ60VV 6QC3^avov i^cb tov avbg oft/xa- 
tog ^kattov iiiibixvkCvSQOv 6(p%"ri6ataL. 

i6ta) y&Q xvXCvSqov roi) TcaQl tijv ^a6iv xvxkov 

20 xavtQov tb JK, xal &7cb tov N ^ii^iatog i^xd^cj iTcl ro K 
il NK^ xal Sl& tov K 7CQbg ^Qd^&g aitfj ijx^co i^ -BF, 
TCaQl Sl tijv KN xvxkog yayQa^pd^cD^ xal i7ca^avx^(*i6av 
aC NZ^ ZK^ Njd^ jdK. oixovv ^Qd^al at TCQbg totg 
Z, ^' xa%^ %v aQa i^paictovtaL al ZN^ N^^ xal a% ya 

25 &7cb tov N '6^^atog (paQ6^avaL &xtLvag xat& t&g NZ^ 
Njd 7ca6ovvtaL' &6ts tb ZA^d ^6vov 6(p%^ifi6^aL. &XI& 
tb ZAjd lkatt6v i6tL tov FAB rj^LXvxh^ov' ro aQa 
ZAA Ha66ov iiiiLXvxkCov ^^pd^Tj^ataL^ tovta6tLv 6 xvIlv- 

4. insid^qTtSQ — 6. ^LoiiistQog] mut. m. rec. in iTtsidri iXae- 
Gtov iotiv 71 BT rijg Sichlsxqov r?}s e^palQccg V. 19. tov'] corr 



OPTICORUM RECENSIO THEONIS. 



189 



culum ZHN efficiat. ducantur autem ab oculis B, F 
rectae BZ, FN in uno puncto contingentes et inter 
se concidant in 0; concident enim, quoniam FB et 
diametrus sphaerae inaequales sunt. itaque rectae a 
puncto ® ad sphaeram adcidentes minus hemisphaerio 
comprehendent. quare ZHN minus hemisphaerio est. 
ergo quae a -B, JT oculis cemitur pars, minor est hemi- 
sphaerio. 

2<S. 

Quomodocunque cylindrus ab uno oculo cemitur, 
minus semicylindro cemetur. 

sit enim K centmm cir- 
culi basim cylindri com- 
prehendentis, et ab oculo N 
ad K ducatur NK, et per K 
ad eam perpendicularis du-' 
catur BF, circum KN autem 
circulus describatur, ducan- 
turque NZ, ZK, N^, ^K 
recti igitur sunt anguU ad 
Z, ^ positi. quare ZN, N^ 
in uno puncto contingunt, 
et radii, qui ab N oculo 
femntur, per NZ, N^ cadent. 
quare ZA^ arcus solus cemetur. uerum ZA/J minor 
est semicirculo FAB, itaque ZA^ minor semicirculo 
cemetur, hoc est cylindms ipse; nam per totam super- 




ex x6 m. 2 V, om. p. itBQi] nocQd comp. p. 22. nsQL] 

Ttagcc comp. p. 27. ictiv V. 28. rifii%v%Xlov] ijfii%vXlvdQOv 
V, fortasse recte. 



190 OPTICORUM RECENSIO THEONIS. 

Sqoq' 6[ioi(og y&g r^ ^&6bi %axiL jc&6av iTtitpdvBiav 
rov xvXCvSqov SsCioiLBV. &6rB ZXov rov zvXcvSqov 
rov fi[i(6Bog iXarrov (paCvsrai, 



5 Tov 8\ &[i[iarog iyytov rBd^svrog rov xvXCvSqov 
iXa66ov ft^i/ i6rac rb 7tBQLXa[iPav6[UVOv imh r&v &itfB(ov 
rov xvXCvSqov^ Sd^Bt. dh fiBt^ov iQ&^d^at. 

i6r(o yaQ xvXCvSqov rov ^bqI rijv pd6LV x^vxXov 
XBvrQOv rb JC, xal aitb rov B 6[i[iarog btcI rb K Ttiv- 

10 rQOV inBiBvid^G} fi BK^ SlA Sl rov JiL ^Qbg d^d^ag i^xd^ca 

ij r^^ xal tcbqI riiv KB x^ixkog yByQaq^d^a)^ xal ixs- 

iBiix^^m^av ai BN^ NK^ BA^ AK Sua Sii rdc 7CQ6rBQ0V 

rb AZN ikarrdv i6riv fiiiixvxUov^ xal 6[iOL(og rfj ^d^Bc 

^okov roi) xvXCvSqov ikarrov t) ro i]^L6v bQad^i^^Brac» 

16 nQo67lx^(o Sii ro Sfifta xal i6rc3 rb O, xal jcbqI ri^v 
0K xvxkog yByQdffd^co^ xal i7CBlBv%%^(o6av al OP, PK^ 
K2J^ 2JQ, oixovv aC aTcb rov O dxrlvBg jCQo67cC7crov6aL 
xard rdg QP^ 0U 7Cs6ovvraL^ aC Si ys dnb rov B 
xard rdg BA^ BN' [iBt^ov ccQa ro NZA rov PZZ. 

20 SoxBi S\ [iBL^ov (paCvB^d^av ro PZ2J rov NZA' ^bC^cov 
yaQ Yi O yc3vCa rijg B yavCag. &6rB xal rov xv- 
XCvSqov iAarrov iiBQog 6(pd^7^6Brac ^ Soxbv S^ ^bv^ov 
bQ&^d^aL, 



1. iTtKpdvLccv V. 2. Ante Ssl^oiisv ins. ro avrb Gvfi§atvov 
m. rec. V. 3. rjiiiGsag V, sed corr. 5. Ss] del. m. rec. V. 
MyYSiov V. 9. K (alt.)] e corr. m. 1 V. 12. Sid — 14. 
6(foc^r]astaL] mg. m. 2 V, om. v. 14. if] om. p. 20. Post 
NZA ras. 1 litt. V. iisttov v. 21. Ante S ins. TtQog rc5 
m, rec. V. Ante B ins. ngbg rw m. rec. V. 



OPTICOEUM BBCENSIO THBONIS. 



191 



ficiem cylindri eandem Tationem exstare demonstrabi- 
mas, quae in basi. ergo e toto cyliudro minus di- 
midio adparet. 



Oculo autem cylindro adpropinquante pars cylindri, 
quae radiia comprehenditur, minor erit, maior autem 

pars cemi uidebitur, 

sit enim K centnim circuli basim cylindri com- 
et ab oculo B ad centrum K ducatur 
BK, per K autem 
perpendicularia duca- 
tur Fj:!, et circum 
KB circulus descri- 
batur , dueantnrque 
BN, NK, BA, AK. 
itaque propter ea, 
quae antea dicta sunt, 
AZN minor est semi- 
circulo, et eadem rsr 
tione, qua ex basi, 
etiam e toto cylindro 
minus dimidio ceme- 
tur. iam oculus ad- 
propinquet et sit *, 
circum O K autem 

circulu9 describator, ducanturque ^P, PK, KS, 2^. 

itaque radii a * adcidentes per 4>P, ^22 cadent, qni 

autem aB adcidunt,perS^, BJV. quaxe iV"Z^>PZ2:. 

uideiur autem PZE maius adparere quam N2,A\ 

nam /. * > B. ergo pars miuor cylindri cemetur, 

uidetur autem cemi maior. 




192 OPTICOEUM RECENSIO THEONIS. 

Kiovov xiixXov ix^vtog tiiv fi&6iv 'bjch tov ivbg 

^lilicctog dQCDiiivov iXa66ov fnLiTCGivCov 6g)d^6staL. 

i6t(o y&Q x(ovov Pcc6ig xvxXog^ o5 xivtQOV tb K^ 

5 xal &7cb tov B Siiiiatog i^x^^ ^^^ ^^ xivtQOv ii BK^ 

xal 8i& tov K TCQbg dQd^ag tfi KB ^ NA^ tcsqI Sh 

tiiv KB xvxXog ysyQA^pd^G)^ xal i7Cs^svx^co6av at BZ^ 

. ZKj jB^, ^K. oixovv dQd^av sl6iv at ucQbg tolg Z, ^ 

ycovcac xad'^ iV aQa i(pd%tovt ai al B^^ BZ^ xal al 

10 ys aTcb tov &iiiiatog axttvsg 7CQ067CC7Ctov6aL xata tccg 

B^y BZ 7Cs6ovvtav. i6tac Sii bQfo^svov tb ZP^ 

ika66ov ov rov NPA. aXl& tb NPA fifiLxvxlidv i6riv' 

tb ccQa ZPA iXa666v i6tcv inLvxvxXCov. &6ts xal tb 

bQcifisvov tov xthvov iXa666v i6tvv '^fivxovCov bfioCcag 

15 ydcQ xal iTcl tcbv XovTCcbv xvxXgjv t&v iv tfj tov xAvov 

iTCVfpavsCa SsC^oiisv. 

Xa . 

Tov S\ ^iiiiatog iyyvov ^statsd^ivtog iv rp avtp 
iicvTciScj) iXa66ov fihv i6tav tb iTcb tcbv Hxlfscov tcsqv- 
20 Xaii^av^fisvov ^iQog^ S6^sv Sh [isv^ov bQcc^d^av. 

i6tc3 yccQ xcjvov ^d6vg x^ixXog^ oh xivtQOv i6ta> 

tb jfiC, Oftfta S^ i6tc3 tb A^ xal ccTcb tov A i%l tb K 

ixslsvx^^ 'h ^^5 ^^^ TCQbg dQd^dg ai)tri ^^-O-o Svdc tov K 

ri FKB^ ysyQd^pd^cj Sh tcsqv ti^v AK xiixXog^ xal ijcs- 

25 g£v%'9'(o^ai/ at AZ^ ZK^ AA^ AK. [istaxsC^d^cj Si^ 

8. z/] J mg thiltivkXov v. 9. BZ] corr. ex z/Z m. 1 V. 
10. Post tov ins. B m. rec. V. 11. ZPz/] Zz/ v. 12. 

NPA (alt.)] N postea ins. V. ictv Vp. 13. ijtiiytvnXiov'] 
pr. X in ras. V. 15. iv rj] in ras. m. 1 V. 18. $s] del. 
m. rec. V. ^yysvov Y, sed corr. m. rec. 22. ini] in ras. 
m. 1 V. 23. iTtiSsvxd^to V, sed corr. 24. rXB] KTB V. 



OPTICORUM RECENSIO THEONIS. 



193 



30. 

Ubi conus circulum basim habens ab uno oculo 

cernitur, minus hemiconio cemetur. 

sit enim circulus, cuius centrum Ky basis coni, et 

a B oculo ad centrum ducatur BK, et per K ad KB 

perpendicularis NA, circum 
KB autem circulus describa- 
tur, ducanturque BZy ZKy 
B^j ^K anguli igitur ad 
Z, ^ positi recti sunt; quare 
B^j BZ in. uno puncto con- 
tingunt, et radii ab oculo 
adcidentes per B^, BZ ca- 
dent. cemetur igitur ZPjdl, 
quod minus est quam NPA. 
uemm NPA semicirculus est. 
itaque ZPA semicirculo mi- 
nus est. ergo etiam ea 

pars coni, quae cemitur, hemiconio minor est; idem 

enim etiam de ceteris circulis superficiei coni demon- 

strabimus. 




31. 

. Oculo autem in eodem plano in locum propiorem 
transposito pars a radiis comprehensa minor erit^ 
uidebitur autem maior pars cemi. 

sit enim circulus, cuius centrum sit jK, basis coni^ 
oculus autem ^ii Ay et ab -/^ ad iT ducatur -/^jK", et 
ad eam perpendicularis per K ducatur FKB, circum 
AK autem circulus describatur, ducanturque AZ, 
ZKy AAy AK iam oculus ^ ad N \3t«a5ss^^scs!S«at 

EuclideB, edd. Heiberg et Meiige. TIL. "^ 



194 OPnCORUM BEC£NSIO THEONIS. 

rb A Siiiia inl tb JV, Tcal tcsqI tiiv KN xiixXog ya- 
yQdtpd^io^ xal in^st^iixd^ai^av at NP^ PK, NU^ ZK. 
(ydxovv aC &jcb tov A ofiiiatog &7tttv6g %Q067cCjttov6av 
ocat& t&g A^^ AZ 7C66ovvtai' &6t6 ^av6ttav tb ZOA. 
6 Svk t& aiftcc 8ii xal at oacb tov N (iiiiatog axttveg 
XQo67C(jctov6ai xatit ticg NP^ N2J 7C66ovvtar 6g)d^66tac^ 
&Qa tb P02, ii6t^ov Sh tb ZOJ tov POU. (paCv6taL 
8% ila66ov' ii6£^(ov yd^Q ^ XQbg rw N yavCa rijg ^r^^o^ 
r^ A ycovCag, 

10 A/3'. 

K(hvov xiixXov i^ovtog ti^v ^&6iv^ i&v iacb tG)v 
6vva(p&v t&v &3cb tov bfifiatog jCQbg tijv tov xdivov 
§d6LV XQo67CC7Ctov6&v &xtCv(ov ^id^^tac Scaxd^&^v Sl& 
trjg i7CC(pav6Cag tfjg tov x(hvov TCQbg tijv xoQv^pi^v aitov^ 

16 Si& Sh t&v &x^6i6&v xal t&v &7cb tov fi^iiatog TCQbg 
ti^v P&6CV tov x(ovov 7CQo67CL7Ctov6G)v iTCiTC^Sa ixpXrjd^y 
iscl Si tTjg xovvrjg roft^g t&v ija^ciS^ov tb Sftfto; t^d^fiy 
tb bQ(0[L6vov tov X(ovov t6ov Sl& TCavtbg d^pd^il^etaL 
t7]g '6il}6(og i7cl TtaQakkrjlov i^CLxiSov t& 7CQ0V7C0X6L[iivcii 

20 i7CL7ciS(p 'b7CaQ%0'66Yig. 

i6t(o y&Q x&vog^ oi ^&6Lg ^iv 6 FA xrixlog^ xo- 
Qv^p'^ Sh tb B 6rifi6tov^ 0^1 fia Sh ro K^ «9?' ov 7Cqo6- 
7CL7Ctit(o6av &xttv6g at KA^ KF a7tt6ii6vaL xat& t& 
P5 A^ xal iTC^^^^^x^cj^av &7cb tcbv A^ F ^rjii^CcDv i^cl 

26 ti^v xoQv^pijv tov xcjvov at AB^ FB^ xal Sl& [ihv tcbv 



6. xal al] corr. ex vLai m. 2 V. 7. Z^J] Z^A pv et 

e corr. V. 8. ^bI^ov v. 9. tc5] rd pv. 11. %6vov V, 

sed corr. 14. iititpaviag v. 15. &%%'bi6&v'\ -£t- e corr. V. 
16. Ante inijesda ras. 2 litt. V. 19. ytaQCiXXrjXov] comp. 

Vpv, omiiibus litteris scriptum add. m. rec. V. 24. ^y P] 
P, J p. 25. al] in ras. V. 



OPTICORUM RECENSIO THEONIS. 



195 



et circum KN circulus 




describatur , ducanturque 
NP, PK, NS, 2JK 
itaque radii ab A 
oculo adcidentes per 
A^y AZ cadent; cer- 
netur igitur Z^A, 
eadem de causa etiam 
radii ab N oculo ad- 
cidentes per NP, NU 
cadent; cemetur igi- 
tur P02J. uerum 
Z0^>P02]. 
uidetur autem minus 
esse; nam 

LN> A. 



32. 

Si in cono circulum basim habenti a punctis con- 
tactus radiorum ab oculo ad basim coni adcidentium 
per superficiem coni ad uerticem rectae ducuntur, et 
per rectas ita ductas radiosque ab oculo ad basim 
coni adcidentes plana ducuntur, oculusque in communi 
planorum sectione coUocatur, pars coni, quae cemitur, 
semper eadem manebit, si uisus per planum plano ab 
initio supposito paraUelum egreditur. 

sit enim conus, cuius basis sit circulus T^, uertex 
autem B punctum, oculus uero sit JST, a quo adcidant 
radii K^j KF in punctis F, /1 tangentes, et a punctis 
^, r ad uerticem coni ducantur AB^ FB^ per FB, FK 
autem planum ducatur, et per ^ 5, jdlK 8imilii«x 5iis!5A. 



196 OPnCOKUM RECENSIO THEONIS. 

r5, FK iTciTcedov ixpsfiXrj^d^io^ Sl& Sh t&v JB^ ^K 

•* OfWLCog atSQOv iTthceSov iK^B^Xiq6%'(o. oimovv ffv[ir- 
7C666trai \r& iTCVTCaSa]' av te yccQ FB^ ^B 6viMlmov6L 
xal al FK^ K^. 6v[uamit(o6av oiv tit iicCTCeSa^ ocal 

j 5 i6t(o ait&v Tcotvii rofti) '^ BK Xeyfo^ 5rt, ojcov av 
hcX f^g BK ted^ tb o^iia^ t6ov tov tcAvov ro oQfo- 
[levov (paCvetai. 

xeC^d^cj yccQ i%l tijg BK tb Z oiiiia^ xal ^O-o 
Sicc tov Z icaQic fihv ti^v K^d ^ ZN^ jcaQcc Sh tijv 

10 FK 7} Z2J, oiwovv at ZN^ ZZ tflg tov x(ovov 
iiCKpaveCag xatcc tk N^ U i^pdTCtovtar tcc yaQ iv 
tfj B r^ tov x(ovov iicupaveCcc t&v TcaQaXXifiXcov 
xiixXcov t[i7Jiiata Siiovd i6tiv. ta aga iv t^ B^F 
tov xcovov iTCLCpaveCa Sia6ti/i^ata bQciiieva t6a cpaCvetav. 

15 iTCel y&Q t6ri i6tCv^ r^v 7ceQiexov6vv at Z2J, ZN^ ycjvCa 

ty TCeQcexoiievr} ijcb tcbv K^^ -K^A ^*^^^ ^^ cpaCvocto 

tb 2JN Sv(li6trjiia tov xcjvov t& ^F Sia^tTliiati. &6d'^ 

" 07C0V av tb ofiiia tedij i%l tfjg KB eifd^eCag^ t6ov ael 

' ' (paveltai tb 6Q(oiievov. 

20 Xy\ 

"l6ov 8% ael tov (fi[iatog aTcb tov X(bvov ajcexovtog 
[lete^oQOv [ihv tov o^iiatog te^ivtog iXa66ov (paCvetai 
tov xiovov tb 6Q(bitevov^ taTcevvoteQov Se (let^ov. 

^6t(o yicQ x(ovov xoQV(pii [ihv TCQbg t& ^ ^rjiieCo)^ 
25 pd6Lg Se 6 BF xvxXog^ xal ^^-O-ca ^ K® TCaQa tijv B/d^ 

1. ^xf/JXtJff-^-co, supra scr. ^, V. 3. ra iitiTtBSa] supra 

scr. V, renou. m. rec. 5. av\ $' ccv Vvp. 8. Z] postea 

ins. m. 1 V. 9. Supra ytccQa (pr.) add. ijtoL TtccQdXXriXog m. 

rec. V. Supra Ttocgd (alt.) add. TtuQdXXriXog m. rec. V. 10. 
FK] in ras. V. 13. iatt p. 16. Krj KN p. 17. i:©] 
corr. ex t6 m. 1 V. 18. av] corr. ex a m. 2 V. 25. oj 

6 nBQl trjv V. 



OPTICORUM RECEN8I0 THEONIS. 



197 



planmn. concident igitur; nam et FBj ^B ei FK, ILd 
concidunt. concidant igitur plana^ et conmmnis eorum 
sectio sit BK, dico, ubicumque oculus in jBX" ponatur, 
partem coni^ quae cematur^ aequalem adparere. 

ponatur enim in 
BK oculus Z, et per 
Z rectae K^ par- 
allela ducatur ZNy 
rectae autem FK par- 
aUela ZS. ZN, ZZ 

igitur superficiem 
coni m Ny Z con- 
tingunt; segmenta 
enim circulorum par- 
allelorum in B F^ 
superficie coni posita 
similia sunt. itaque 
distantiae, quae in B^F superficie coni cemuntur, 
aequales adparent. nam quoniam angulus rectis ZU^ 
ZN comprehensus angulo rectis K^, KF comprehenso 
aequalis est, distantia UN in cono distantiae ^F 
aequaKs adparet. ergo ubicumque oculus in recta KB 
ponitur, pars, quae cemitur, semper aequalis adparebit. 




33. 

Oculo uero semper idem spatium a cono distante, 
si sublimis oculus ponitur, pars coni, quae cemitur, 
minor adparet, si demissior^ maior. 

sit enim coni uertex ad punctum ^, basis autem 
circulus BFj ducaturque K& Teo.t-a.^ Y^A y^^^iSi.^ ^ 



198 OPnCOKUM KECENSIO THEONIS. 

Tcal xsied^o) rb iiiiuc istl rov 0. q>riiil di^ iXa€6ov 
bfpd^^B^d^av rov Ttfovov rb bQAfievov rsd^dvtog rov Oft- 
[laros inl rov & 6riii£iov i^XBQ ixl rov 27. iTCs^siixd^G}' 
6av y&Q &7cb rov A 6t[\iBlov iscl r& ®^ U 6rjii6ta at 
h JS^ ^H xal ixfiefiXii6d'(06av isd ra JV, ji. oixovv 
i^C rs rov N xal ixl rov A 6rjiisLOv rsd^ivrog rov 
oiiimrog avi6a (paivsrac r& bQd)yi,sva rov xiovov^ xal 
iXa66ov ^hv fpaCvsrai rb ngbg rdi JV, fist^ov Sh rb 
xgbg rm A. t6ov Sh rb ^hv TCQbg rp N r3 XQbg rp 0, 
10 rb 61 TCQbg rm A rdi TCQbg r& U^ cjg iv rm ocQb airov 
iSsCx^rj. rov aQa ofiiiarog JCQbg rp & 6rj[isCp ovrog 
iXa66ov (paCvsrai rb bQ(0[LSvov rov X(hvov i^TCSQ sCQbg 
rm U. 

4 

xs\ 

16 ^Ev xvocXp i&v aTcb rov xivrQOv JCQbg dQd^dg rvg 
ax^fj r& roi) xvxXov iTCiTciSG)^ iicl Sh ra^&trjg rsdrj rb 
ofifia^ t6ai al SvaiisrQoi roi) x^^xXov (paCvovrai. 

i6r(o y&Q x^ixXog^ oi xivrQOv rb K^ xal ajcb rov K 
TCQog dQd^ag avif{X^(o r& iTCLTciSa) rov xvxXov fj KB^ 

20 ro Sh o^^a xsC^d^co ijcl rov jB, xal SiAiLsrQOv fjx^^^^'^ 
al FA^ AZ. (prjiil Sij rijv AF rfj AZ t6rjv (paCvs^d^av. 
i7C£^svx^(o6av yccQ at BA^ BZ^ BF^ BA. obxovv Svo 
at BKy KZ Sv6l ralg BK^ KT l'6av sC^lv ixariQa 



2. 6q)d"riasrat p, d}(pd"i/jastca v. oiQmiisvov v, sed corr. 

3. Z] om. V. 5. A] corr. ex z/ m. 2 V. 9. rc5 (sec.)] 

r6 V. rc5 (tert.)] r<J pv, et V, corr. m. rec. 10. rm A 

rc5] ro A ro v. rc5 (tert.)] r(J pv. 11. rc5] ro v. ari- 

lisiov V, et V, sed corr. ovrag v. 12. iXoiacav V, sed 

corr. 15. &7t6 rov liSvrQov] in ras. m. 1 V. 19. rc5] r6 v. 

20. rov] om. p. 23. Ante BJf^alt.) eras. T V. ' KF] 

corr. ex Kd m. rec. V. 



OPnCORUM RECENSIO THEONIS. 



199 



oculus in ® ponatur. dico igitur^ partem minorem 
coni cemi oculo in @ posito quam in 2J, ducantur 
enim a ^ puncto ad @, U puncta ^0, ^2J et ad 

Nj A producantur. 
itaque partes coni, 
quae cemuntur, oculo 
in N posito et in ^ 
puncto inaequales ad- 
parent, et quae ab JV 
cemitur, minor ad- 
paret^ maior autem, 
quae ab A cemitur 
[prop. 31]. uerum 
quae ab JV cemitur, 
aequaKs est ei, quae a S cemitur, quae autem 
ab A cemitur, ei, quae ab 27, ut in propositione prae- 
cedenti demonstratum est [prop. 32]. ergo pars coni, 
quae cernitur, minor adparet oculo ad punctum 
posito quam ad 27. 




34. 

Si in circulo a centro ad planum circuli perpendi- 
cularis recta erigitur, in eaque oculus ponitur, dia- 
metri circuli aequales adparent. 

sit enim circulus, cuius centmm sit X", et a JST ad 
planum circuli perpendicularis erigatur KB, oculus 
autem in B ponatur, ducanturque diametri FAy ^Z. 
dico igitur, adparere Ar= AZ, ducantur enim BAy 
BZj Br, BA. itaque duae rectae BK, KZ duabus 
BK, Kr aequales sunt singulae sin^ilia, \i&Tssssi. ^ioissjss^ 



200 



OPTICOKUM RECENSIO THEONIS. 




ixaraQcc. i6xi, S\ Ttal ^ P 

ycDvCa tfi U t6ri' t6ri aga 

xal fi BZ ^d6ig r^ BF 

fid6£c. SiA rct aiytic Si^ xal 
6 ^ BJ rfj BA i6riv 16^1. 

S^o Sii at z/5, BZ Sv6l 

ratg FB, BA t6aL 6l6iv, 

i6rv S% xal ii AZ rri FA 

t6ri' ymvCa &Qa ^ i)Xo ABZ 
10 yavCa rfj inh FBA t6ri 

i6rCv. rk S\ imb t6a)v yco- 

VL&v 6Q(0[i6va t6a cpaCverai. 

t6ri aQa ^ FA rfj AZ (paC- 

varav. 
15 Xa. 

Kal ikv ^i VTtb rov xavrQOv avaxd^et^a fn^ TCQbg 

dQd^ctg ^ ra iTCLjtiSo)^ t6ri Sh rfj ix rov xivtQOV^ t6av 

al SidiLStQOL (paviJ6ovraL, 

i6rc3 xvxkog^ oi xivrQov rb Kj xal djcb rov K ^iii 
20 ncQbg dQd^dg dvijx^^ ^^9 imTtiSG) fi KB^ t6ri S% fiyro 

rri ix rov xivtQov rov x^^xlov^ xal iTts^eiix^co^av djtb 

rov B 6rjii6Cov a[ airal ratg TCQdrsQOv. ovxovv iTtsl 

t6ai dUiXaig sl6lv at AK^ KB^ KZ^ dQd^i^ av strj i^ 

TtSQLSXoii^ivri ycovCa ifTtb r&v ZBA. Svd rcc aira Si^ 
26 xal ij x)7tb ABF dQd-ij dv strj' t6aL aQa i6ovraL dlXi^- 

XaLg. rd Si ys i)7tb t6c3v yfovvmv dQchiisva t6a cpaC- 

vsraL. t6rj aQa ij AZ ry AT cpaCvsraL. 



2. 2] 2 yavia p. 3. BF] corr. ex BJ m. rec. V. 5. 

JBz/] corr. ex BT m. rec. V. 10. FBA] FAB p. 11. 

iaxi p. 17. Post imTtsdo} add. tov yivvlov m. rec. V. 

18. Ante al add. xal ovxoig m. rec. V. 19. Post ^cyrco 



OPnCOKUM RECENSIO THEONIS. 



201 



/. P= 27;^) itaque etiam Br= BZ. eadem de causa 
etiam B^ = BA. itaque duae ^B, BZ duabus FB, 
BA aequales sunt. uerum etiam AZ = ILdf; quare 
L ABZ = FBA. quae autem ab angulis aequaKbus 
cemuntur, aequalia adparent [def. 4]. ergo FA rectae 
AZ aequalis adparet. 

35. 
Etiamsi recta e centro erecta ad planum perpen- 
dicularis non est^ diametri aequales adparebunt^ si 
modo radio aequalis est. 

sit circulus, cuius centrum 
sit Ky et e JST erigatur KB 
ad planum non perpendicu- 
laris, radio autem circuK ae- 
quaKs sit, et a 5 puncto 
eaedem rectae ducantur, quae 
antea. itaque quoniam 
AK=KB = KZ, 
rectus erit L ZBA. eadem 
de causa L ABF rectus erit; 
itaque L ZBA = ABF. quae autem ab anguKs 
aequaKbus cemuntur, aequaKa adparent [def. 4]. ergo 
^Z rectae AF aequaKs adparet. 




1) Litteris P et Z mire significantur anguH BKFy BKZ. 
in figura etiam in angulo BKZ littera T posita est in pv 

(om. V). 



add. yaQ p et m. rec. V. 20. Post iTeinsSo) add. rot) ii6%Xov 
m. rec. V. 24. Supra ZBJ add. B m. rec V. tdl --a 

supra scr. V. 25. xat] om. v. 26. ^^ qtsl, ^, 



202 



OPnCORUM EECENSIO THEONIS. 



^AXX&, 8ii ii AZ (itjtb t6rj i6t(0 rfj ix tov xdvtQOv 

fti^ra TtQog dQd^&g t^ tov Tc^xkov imTciSo)^ t6ag S% ya>- 

vCag noiBCtm t&g ijch AAZ, 

ZAFxal EAZ^ ZAB. kiyo^ 
5 StL xal ovtcog aC SiaiLStQov 

t6av q>av7J6ovtac. iTCsl yicQ 

i:6rj i6tlv fi AA tfj AF^ 

xoLvij Sh rj AZ^ xal ycDvCag 

t6ag 7CBQii%ov6iv^ fia6ig aQa 
10 ii JZ pd6si tri Zr t6ri 

i6tlv xal ycjvCa ^ iTcb ^ZA 

r^ imh AZT. h^oCcog Si^ 

SsC^ofisv^ Sti xal ij vTch 

EZA tfi iTch AZB i6tiv 
15 t6ri. SXrj aQa fj 'bjch AZB oAg tf^ x}7ch EZF i6tiv 

t6rj. &6ts at SidiistQOi t6ai q^avrj^ovtai. 




^Eav S% r] aTch tov o^iiatog TCQhg th xivtQOv jcqo6- 
7cC7Ctov6a tov xvxXov ^T^ts TCQhg dQd^ag fi rp tov xiixXov 

20 i7Ci7ciS(p [i^^ts t6rj fi tfj ix tov xivtQOv ^fjts t6ag ycDvCag 

7CSQii%ov6a [istd tmv ix tov xivtQov^ [isC^cov Sh f^ iXd6- 

6c3v tfjg ix tov xivtQov^ avi60i al Sid[LStQ0i q^avovvtai, 

l6tco ydQ xvxlog^ oi xivtQov ro A^ xa\ aTch tov B 

o^^atog i7cl th xivtQov tov xvxXov svd^sta fjx^ca ij BA 

26 xal i6t(o i^rits TCQhg dQd^ag t& i7Ci7ciSG} lii^ts t6r} tfj 
ix tov xivtQov tov xvxXov ^iTJts t6ag ycavCag 7CSQi- 
i%ov6a fistd t&v ix tov xivtQov. Xiyco^ ort al Sid- 
fistQOi tov xiixXov avi60i (pav7J6ovtai. 

7. tari\ «^<y^ "^- 11- ^<y^t P- ^ZA] EZA p. 14. 

^ZJ] Z e corr. V. 'bno] ccno v. Ib. JZB] JBZ V, corr. 



OPnCOKUM RECENSIO THEONIS. 



203 



lam uero AZ ne sit radio aequalis neue ad pla- 
num circuli perpendicularis, sed efficiat L^AZ = ZAT^ 
EAZ = ZAB. dico, sic quoque diametros aequales 
adparere. nam quoniam AA^AF, et AZ communis, 
aequalesque angulos comprehendunt, erit AZ = ZT 
et L AZA = AZr, similiter demonstrabimus, esse 
etiam LEZA = AZB. itaque totus LAZB = EZr. 
ergo diametri aequales adparebunt. ^) 

36. 
Sin recta ab oculo ad centrum circuli adcidens 

neque ad planum circuli perpendicularis est neque 

radio aequalis neque cum 
radiis angulos aequales com- 
prehendens, sed radio uel 
maior uel minor, diametri 
inaequales adparebunt. 

sit enim circulus, cuius 
centrum sit A, et ab oculo 
B ad centrum circuli recta 
ducatur BA et sit neque 
ad planum perpendicularis 
neque radio circuli aequalis 
neque cum radiis aequales 

angulos comprehendens. dico, diametros circuli in- 

aequales adparere. 




1) Litteras figurae dedi ex Vv, in p nostris ita respondent, 
ut pro ^, B, r, z/, E, Z sint K, Z, A, z/, T, B. 



m. rec. 18. iccv 8s ^] in ras. V. 21. ftijfcov V, sed corr.^ 

flSl^OV V. 



204 OPTICORUM RECENSIO THEONIS. 

rfl AB^ ij Sh ^K avC6ovg ^ovov6a ytoviag nQog rfj 
AB^ Tcal ijcstBiix^co^av at BF^ BJ^ BZ^ BK^ i6r(o 
Sh TtQdrsQOv ii BA rf}g AK ^i^cav. oixovv fiSL^(ov 
5 i6rlv ^ %BQVBxo[Livri ycovva hxo r&v FBZ rfjg tcbqv- 
Bxopbivrig i>7cb r&v KB^^ 6)g iv rotg d^BCjQiliia^vv &Jto- 
SBvxwrav. r& Si ys imb lisv^ovog ycoviag bQihjUva 
pLsCtpva (paCvsrav lisC^ov ccQa i^ FZ rfig dK tpaCvsrav. 
ikv S% fi BA rfjg AK iXA66cov ^, iisCtcov (paCvsrav fi ^K 
10 rfig rZ. 

^E6rc3 3«t;xAoff, o5 ksvxqov rb A^ ^ii[i,a Sh rb B^ aq>* 
ov ij ijtl rbv xvxXov xdd^srog ayoiisvrj iiij xvTcrsra) iitl 
rb xsvrQOv ro A^ cJAA' ixrdg^ xal S6rco ^ jBP, xal ijts- 
^svx^G) aTtb rov F ijtl rb A ii FA^ irv Sh ajtb rov A 

15 iTtl rb B ^ BA. Xiyco^ orv 7ta6&v rcbv Sv& rov A 
Svayofiivcov sid^sv&v xal 7tovov6cbv TtQog rfj BA ycD- 
vCag iXaxC6xri i6rlv ^ vTtb r&v FAB. SvTJxd^co yaQ 
sid^sta ij AAE^ xal ^-O-oj ajtb rov F inl ri^v /lE 
xa^srog iv xp ijtvxiSco ij FZ^ xal iTtB^svx^cD ij BZ' 

20 xal ij BZ ccQa ijtl xi^v AE xdd^sxdg i6xvv. ixsl ohv 
dQd^ii ii iitb rZA^ ij i)7tb AFZ aQa ikd66cov i6xlv 
dQd^fjg' iisC^cjv ccQa ii AF TtlsvQa xfjg AZ. i^ BA ccQa 
TtQog xi^v AZ iisC^ova X6yov sxsv ^itSQ TtQbg xi^v AF. 
«AA' i] vTtb xcbv AFB ycjvCa xal ij vjtb x&v BZA 

25 sl6vv dQd^aC^ xaC sl6vv ai FA^ AZ avv6ov* xal lovTtrj 

4. (isit(ov (pr.)] fislSov v. iisitonv (alt.)] iisT^ov v, (isl- in 
ras. V. 6. Post totg add. 7tQ0tSQ0i>g m. rec. V. &7to- 

SsUvvtaC] mut. in &'3to8s8siv.tai m. rec. V. 7. (isL^cDvog v, 

sed corr. 11. >LJ' Vpv. ytsvtQov] m. rec. V, comp. m. 1. 
12. &yoDiisvri V, sed corr. 16. noiova&v] -a&v e corr. m. 

rec. V. Post ty ras. 1 litt. V. 17. t&v] del. m. rec. V, 
seg. ras. 2 litt. v. 18. tTJv] to v. 22. iisl^ov v. AF] 



OPTICOKUM RECENSIO THEONIS. 



205 



ducatur enim diametrus FZ 2idi AB perpendicu- 
laris, ^K autem cum AB angulos inaequales efficiens^ 
ducanturque BF, B^, BZ, jBJST; sit autem prius 
BA > AK. itaque L FBZ > KBA^ ut in propositio- 
nibus^) demonstrabitur. quae autem ab angulo maiore 
cernuntur, maiora adparent. itaque FZ maior ad- 
paret quam AK. sin est BA < AK^ AK maior ad- 
paret quam PZ. 

Sit circulus, cuius centrum sit Ay oculus autem 
sit B, a quo quae ad circulum perpendicularis ducitur, 
in A centrum ne cadat, sed extra, sitque jBP, et 

ducatur a T ad ^ recta 
FAy praetereaque ab A 
ad jB recta BA. dico, 
omnium rectarum, quae 
per A ducantur et cum 
BA angulos efficiant, 
minimum angulum ef- 
ficere rA, scilicet 
L FAB. ducatur enim 
recta AAE, et a F ad 
AE perpendicularis in 
plano ducatur FZ, et ducatur BZ\ itaque etiam BZ Sid 
/dE perpendicularis est. iam quoniam L TZA rectus 
est, L ATZ minor est recto; quare Ar>AZ. itaque 
BA:AZ>BA: AR anguli autem ATB et BZA , 

1) Significantur lemmata, quae sequuntur. 




corr. ex AB v. 24. t&v (utrumque)] del. m. rec. V. 86. 
FA] r in ras. V. 



206 opncoHUM becensio theonis. 

aQa fj iycb t&v ZAB r^g ino t&v FAB i6tt iisi^cav. 
dfioicog Sii Ssixdii^stai^ Sti Tcal 3Ca6&v t&v 8i& tov A 
Siayo\jdvfQv sid^et&v xal noiov6&v nqhg ty AB aid^sia 
ycavCav iXa%C6tri i6tlv ij imb x&v FAB. 

6 TOrt ij ZB tri AE i6ti sCQbg dQd^dg^ SeC^oiiev ovt(og, 

iycsl 'fi BF t^ tov tcAkXov imTciSp i6tl sCQbg dQd^dg^ 

xal scdvta aqa tk Sik r^g BF iicCneSa ixPaXXdfUva 

tp tov xihckov iTCLuciSc} i6tl sCQbg d^d^dg. l^v Sh t&v 

Sio^ trig Br ixPaXXoiiivcDV imniSoiv i6tl tb BFZ 

10 tqCycavov' xal tb BFZ aqa tqCycovov t^ tov xihcXov 
ixLxiSG) i6tl TCQbg dgd^dg. iicsl oiv Siio inCnsSa t6 ts 
tov EA xiixXov xal tb tov BFZ tQiyAvov tifivov6tv 
akkr^kc^ xal tfj xotv^ ait&v tofifj tfj FZ XQbg dgd^dg 
i6tiv fi ZA iv t& tov xvxXov iTCVTciSc)' Tcdd^stog yctQ 

16 ^xtai fj rZ ijcl tiiv EA' xal ii ZA aQa tp tov BFZ 
tQtyavov inL^iSca i6tl ngbg dgd^dg, &6ts xal ngbg 
7cd6ag t&g ccTCtofiivag avtijg si^sCag xal ov6ag iv tm 
tov rZB tQty6vov intniSci i6tl uCQbg dgd^dg' ii AZ 
ccQa tfi ZB i6tt yCQbg dgd^dg, dvdnaXtv &Qa ii BZ 

20 tfi EZA Sta^itQp i6tl jCQbg d^d^dg. 

"E6tGi Siio tgCycova tk BTA^ BZA dQd^ctg i%ovta 
tctg ycQbg tovg F^ Z ycovCag^ xal ^ BA sCQbg ZA (isC- 
^ova k6yov i%sto ^xsq TCQbg ti^v FA. kiycj^ otv (uC- 
^(ov i6tlv ii i)7cb ZAB ycovCa tijg i)7cb FAB ycovCag. 
26 ixsl yaQ ij BA TCQbg ti^v ZA ^sCtpva k6yov i%st ^tcsq 
TCQbg tijv FA^ xal dvdTCakvv ccQa ij ZA TCQbg ti^v AB 



1. t&v (utrmnque)] del. m. rec. V. iativ Vv. 3. Post 
Tj ras. 1 litt. V. 4. t&v] del. m. rec. V. 5. ^tj' Vpv, 

del. in v. Post 2frt ins. ds m. rec. V. iativ Vv. 6. 

^ar/p Vv. 8. tobv] corr. ex t& m. 2 V. 9. i7ipccX7L6(tsvov 



pPnCOHUM HECENSIO THEONIS. 207 

recti sunt, et FA^ AZ inaequaJes. itaque etiam 
L ZAB > FAB, simiKter demonstrabimus^ omnium 
rectarum, quae per A ducantur et cum recta AB 
angulos efficiant, minimum angulum efficere FA, 
scilicet L FAB. 

ZB ad AE perpendicularem esse, sic demonstra- 
bimus: 

quoniam ^r ad planum circuli perpendicularis est, 
etiam omnia plana^ quae per BF ducuntur^ ad pla- 
num circuli perpendiciilaria snnt. inter plana autem 
per Br ducta etiam triangulus BFZ est; quare etiam 
triangulus BFZ ad planum circuli perpendicularis est. 
iam quoniam duo plana^ et circuli EA et trianguli 
BFZ, inter se secant, et ad FZ communem eorum 
sectionem perpendicularis est ZA in plano circuli 
(PZ enim ad EA perpendicularis ducta est), ZA 
etiam ad planum trianguli BFZ perpendicularis est. 
quare etiam ad omnes rectas eam tangentes et in 
plano trianguli FZB positas perpendicularis est. ita- 
que AZ ad ZB perpendicularis est. ergo uicissim 
BZ ad EZA diametrum perpendicularis est. 

Sint duo trianguli BFA, BZA angulos ad 1] Z 
positos rectos habentes, et sit BA : ZA > BA : FA. 
dico, esse L ZAB > FAB. nam quoniam est 

BA:ZA>BA:PA, 



ininESov V, corr. m. rec. iexiv Vv. 11. iaxiv Vv. 14. 
Tc5 xov] -& r- in ras. V. 16. t©] x6 v. 16. iaxiv Vv. 

18. XQiymvon V, -ov in ras. V. iexiv V. JZ] JE v, 

19. iaxiv Vv. 20. iaxiv Vv. 21. l&' Vpv, in v di 
22. Ante ZA ins. xi/jv m. rec. V. 23. iisi^mv'] iistiov v. 25. 

ydo] om. v. 26. FA] A e corr. V. ilB\ A <^ <iCTct..^. 




208 OPTICORUM RECENSIO THEONIS. 

iXA66ovu kdyov bxbl^ oi i%ev ^ FA nqog AB' &6ts 
i5 FA %QhQ AB lui^ova kdyov ixai fptBQ ii ZA stQog 
AB. 7t£7CotiJ6d'C3 cSv, d)g fi FA 
TtQog ABy ovt(og 'fj ZA TtQog 
6 ikd66ova tfjg AB ti^v AA' 
leoyAvLcc aqa i6tl tcc tQ^yava 
tcc BFA^ AZA. &6ts t6ri 
i6tlv ii ino FAB ymvCa tQ 
ino ZAA. fisi^cov aqa fj inh 
10 ZAB ycovia tijg iTtb FAB. 

"E6tco xihcXog 6 AFBA^ xal Si^^xd^cj^av di5o Sid- 
fistQOL aC AB^ FA ts^vov^av &klifjlag itQog d^d^dg^ 
bfifuc Ss i6tci) ro E^ &(p' o5 i^ ^^^ ^^ TcivtQOv 6r^- 
^svyvvfiivrj ij EZ TtQog dQd^&g fihv i6t(o tfj FAj TtQbg 

15 (Jf tijv AB tvxov6av ycaviav JtSQisxitco^ Tcal i6t(o ^i 
EZ sxatiQag t&v ix tov xivtQov fisc^cav, ixsl oiv 
^l FA sxatiQcc tcbv AB^ EZ i6tc TtQog d^d^dg^ xal 
Ttdvta aQa td Sid tfig FA ijtiTtsSa ix^akkdiisva r© 
S(,d tG)v EZ^ AB imjtiSp jtQog d^d^dg i6tiv. fjx^^ 

20 oiv ditb tov E 6rj^siov iitl tb ijtoxsLfisvov ijtCjtsSov 
xdd^stog' i%l t^v xolv^v ocQa tofiijv nCntsi tcbv im- 
TtiScav tijv AB. nLittitG) ohv xal i6tc3 ij EK^ xal 
Snfix^cn SidiistQog ij H&^ xal xsC^^^co tfj Sva^itQp tov 
xvxlov l'6ri r] AM xal tstiiiifj6%^co SC^a xatd tb N^ 



3. itsitoiBleQ^co V. 5. AJI corr. ex ^B m. 1 V. 6. 

iativ Vv. 7. BrA] A corr. ex z/ m. rec. V. 9. ilbI^ov v. 
10. ZAB'] B e corr. m. rec. V. 11. \l' Vpv, del. v. Ante 
6vo eras. al V. 17. icxLv Vv. 20. Post arnieiov add. in 
media linea — Vv. 23. HS] corr. ex E0 V. 24. Post 
^M del. TfQbg dg&dg p. 



OPTICOEUM RECEN8I0 THEONIS. 



209 



etiam e contrario est XA : AR < TA : A&. quare 
rA:AB>ZA:A&. fiat igitur TA\AR-^ZA:A^, 
quae minor est quam AR. itaque trianguli BTA, 
^ZA aequianguli sont; quare i TAB = ZAA. ergo 
L ZAB > TAB. 

Sit circulus ATBJf et ducantur duae diametri 
AB, TA inter se ad angulos rectos secantes, oculus 
autem sit E, a quo quae ad centmm ducitur EZ, 
ad TA perpendicularis sit, cum AB autem quemuis 




angulum contineat, et EZ utroque radio maior sit. 
iam quoniam T^ ad utramque AB, EZ perpendicu- 
laris est, etiam omuia plana, quae per T^ ducuntnr, 
ad plauum per EZ, AB ductum perpendicularia aunt. 
iam ab E puncto ad planum subiacens petpeudicataris 
ducatur; cadit igitur in AB commvmem. ^>a:&Kic<:£i^ 

EneJidBB, odd. Hdberg alMmiBa. ■TO. Vk. 



210 OPTICORUM RECENSIO THEONIS. 

xal dvTJx^fo &7tb tov N tfj AM TtQog dQd^ag fi€tsa)Qog 
sid^eta ii NS^ xal i6t(o ij NS tfj EZ t^rj' tb &Qa 
tisqI f^v AM yQag}6fisvov t^fl^a xal i^xd^svov 8i& 
tov S fislidv i6tiv 'fi^vxvxXiov^ i%siSifinsQ fi NS yLsCt,(ov 
6 i6tlv sxatSQag t&v AN^ NM. i6t(o tb AUSM^ xal 
iTCs^s^^xd^cjeav aC SA^ SM. fj &Qa Jt^bg tp S ycjvca 
'fj TtSQtsxo^svrj i)7tb t&v ASM Cerj i6tl tfj JtQog rp E 
6yi^sCg) tfi TtSQLSxo^svrj {)7tb t&v i7tit,svyvvov6G}v tb E 
Tcal t&. f, ^ 6ri^sta. ixxsCcd^co tfj i^tb tcbv EZ^ ZH 

10 Cerj fj {)7tb t&v AN^ NO^ oial a^prjQri^d^co l'6ri tfj EZ 
fl NO^ xal i7ts^svx^co6av aC AO^ MO^ xal 7Csqi- 
ysyQafpd^G} 7tsql tb AOM tgCycovov tfiflfia xvxkov tb 
AOM. s6tai (Jt) Tial 'fj 7tQbg ta O ^rj^sCa) ycovCa t6rj 
tfi {)7tb tcbv HES. iti xsCad^co tfj {)7tb tcbv EZK Cffrj 

15 ^ i7tb tcbv ANU^ xal ixxsC^d^cj tfj EZ l'6ri fj Nlly 
xal i^ts^stix^c^eav at AH^ HM^ xal ^tSQLysyQccg^d^cy 
TtsQl tb AHM tQCycjvov tfi^fj^a xvxlov. i6tai Sij 
xal 'fj TtQbg rp H ^rjfisCG) l'6rj tfj 'b^tb AEB ycovCcc. 
i7tsl ovv ^sC^cjv i6tlv 'fj 7tQbg rco S tfjg 7tQbg tip O 

20 ycovCag' 'fj ^hv yccQ 7tQbg t(p S l'6rj i6tl tfj 7tQbg rc5 27 
ycovCa^ 'fj Sh 7tQbg rp 2J fisC^ov i6tl tfjg 7tQbg rc3 O 
ycovCag' tQcycovov yo:Q tox) AUO ixtog i6tLV' xal 'fj 
7tQbg t(p S &Qa ^sC^CDV i6tl tfjg TtQbg t(p O' xaC i6tLV 
'fj fiiv 7tQbg rp IS l'6rj tfj 'i)7tb FE/l^ 'fj Ss 7tQbg rp O 

25 tfj 'l)7tb HE&^ ^sC^cjv &Qa cpavtj^staL xal 'fj F^ tfjg 
HS. 7tcchv fj ^lv 7tQbg t(p O ycjvCa tfj 'VTtb HES 



3. ccQxoiisvov V, corr. m. rec. 4. ilsI^cov] ftstfov v. 6. 
ISIA] Z^ p. rc5] in ras. V, ro v. 7. iari.v Yv. 9. iy,- 
7isLa&a)] ht ■iislad'0} e corr. p. ZH] H e corr. v. 11. AO] 
O e corr. v. MO] corr. ex M@ v. nsQLyQdtp&a V, sed 
corr. 12. Post t6 (pr.) 1 litt. eras. v. 13. AOM] O e corr. v. 



OPTICORUM RECENSIO THEONIS. 211 

sectionein. cadat igitur et sit EK, ducaturque dia- 
metrus H&^ et diametro circuli aequaJis ponatur AM 
seceturque in iV in duas partes aequales, et ab N 
ad AM perpendicularis subUmis erigatur recta NS^ 
sitque NS = EZ, segmentum igitur circum AM 
descriptum et per IS ueniens maius est semicirculo, 
quoniam NS utraque AN, NM maior est. sit AUSM, 
ducanturque SAy SM. itaque angulus ad S positus, 
qui rectis AS^ SM comprehenditur, angulo ad E 
punctum posito, qui rectis ab E a.d F, ^ puncta 
ductis comprehenditur, aequalis est. ponatur 

L ANO = EZH, 

sumaturque NO = EZj et ducantur AOy MO, et 
circum triangulum ^OM describatur segmentum cir- 
culi AOM, erit igitur etiam angulus ad O punctum 
positus angulo HE S aequalis. praeterea ponatur 
L ANn=EZK et Nn=EZ, ducanturque Ally 
TLMy et circum triangulum AJJM describatur segmen- 
tum circuli. erit igitur etiam angulus ad JT punctum 
positus angulo AEB aequalis. iam quoniam L S> O 
(nam LS= ^, sed L ^> O, quia angulus externus 
est trianguli A2J0'^ itaque etiam LS>0), uerum 
L S = FEJ , L O == HE&, maior adparebit 
Tz/ quam H® [def. 4]. rursus LO = HE@, 



rc5] T(J V, et V, corr. m. rec. 18. rc5] t6 v. AES] 

corr. ex AEB m. rec. V. 19. iistiov v. rw (pr.)] r6 v. 

rfjg] e corr. V. 20. icrlv Vv. Kccl yccg aii(p6r8QccL ip 

rc5 a^rco rfiiffiart sIgl mg. m. 2 p. 21. iiettov v. 22. iartp, 

23. fiffjoi; V. icrlv Vv. rfjg] corr. ex riji m. rec. V. 

rc5 (alt.)] r6 v. 24. ro5 (utramque)] rd v. 26. fist^ov T. 

26. H@] H e corr. p. rai] r6 v. 



212 OPnCORUM RECEXSIO THEOXIS. 

i6Tiv t<fr^^ ^ dl TCQos ra 11 rij vxb AEB' fi^f^ov ds 
ri O tf^g n, fuC^/(ov &Qa tpavrfiExtu ^ HS rf^g AB 
iv^sCag. 

Mri i6xG} drl luilcjv ^ ascb tov oyLfuxtog ixl xb 

5 xevTQOv imtjEvywiuvri Tf^g ix tov xsvtqov^ aXkii ikd6' 
6<ov' i6Tai dri X£qI Tag dtafLdTQOvg tovvccvtiov ^ yaQ 
t6t6 luCtjav T&v di€cii£TQ<ov vvv ild66<ov <p<£vrfi£xai^ 
7} ds il<i66<ov iuC^<ov. i6T<o xvxkog 6 ABFA^ Tcal di- 
rJXd^cD^av dvo duiiuTQOL t£iiv<yv6cu aXX^^lag XQbg dQ^ag 

.0 a[ AB^ FA^ £T£Qa di Tig dnf^x^o} ri HSj oiiiuc dl tb El, 
a<p ov ij ixl ro Z TcivTQOv ini%£vx^£i6a i6TCi) fi EZ 
i?ji66G)v ov6a ixatiQag t&v ix tov xivTQOv^ nQog 
iQ^icg d% Tfj FA i^Tco fj EZ^ Tcal X£c6^c3 rg rou xv- 
xkov duciUtQC} t6rj rj AM xal T£Tiirfi^a SCxa xccra 

L5 ro iV, xal dvrjx^<o dxb tov N XQbg dQ^dg '^ NS t6rj 
tfi EZ^ Tcal Tt^QLy^yQdcpd^a 7t£Ql trjv AM ocal tb S 
6ri(i£tov t(ifi(ia xvxkov tb ASM' ietai Sij iXa66ov {^11- 
xvxlCov^ i7t£LS7]7t£Q fi NS iXd66G)v i6tl tflg ix tov 
xivTQOv. £6taL Srj fi TtQog rp S 6riii£Ca} ycovCa fj 7t£QL- 

SO Exofiivrj V7tb tcbv ASM t6rj rg TtQog ta E^ 7t£QL' 
£X0fiiv7j Si V7tb ta)v FEA. itL X£C6%G3 rg vTtb tav 
EZH t6rj fi V7tb tav ANO^ xal dcpnQTi^d-cj tfj EZ 
t6ri fj NO^ xal ^tEQLy^yQd^pd^ci) 7t£Ql tfjv AM xal ro O 
6rjii£tov tb AOM tiifjfia. fj Srj, 7tQbg tp O 6rjii£CG} yca- 

J5 vCa fj 7t£QL£xofiivrj VTtb tG)v AOM t6rj i6tl tfj 7tQbg 

tip E tfj 7t£QL£X0llivrj VTtb tG)V SEH. itL X^C^d^G) tfj 

i)7tb t&v AZ^ ZE t6rj fj V7tb tcbv AN^ NH^ xal 

1. ^isl^ov V. 2. ^st^ov V. 3. sv&Siccg] yavlag Y, svd^siag 
ycaviag pv. 4. ft' Vv, fia' p. 7. iisttov v. 11. iiti- 

isvx^fjGcc V. 18. iativ Vv. 19. tc5] t6 v. 22. 17] om. v. 

24. Tfi^^a] riLf](ux kvkXov p. 25. 17] supra scr. m. rec. V. 

iativ Vv. 



OPTICORUM RECENSIO THEONIS. 



213 



L n= AEB, et LO>n. ergo etiam H& maior 
adparebit recta AB [def. 4]. 

lam recta ab oculo ad centrum ducta radio maior 
ne sit, sed minor. tum de diametris contrarium eueniet; 
nam quae diametrus antea maior erat, nunc minor 
adparebit, minor autem maior. sit circulus ABFJ, 
ducanturque duae diametri AB^ Fjd inter se ad angu- 
los rectos secantes, alia autem quaeuis sit H&^ oculus- 

H 

A 




que sit E^ a quo quae ad Z centrum ducitur, sit EZ 
utroque radio minor, perpendicularis autem sit EZ 
ad FAy et ponatur AM diametro circuli aequalis 
seceturque in duas partes aequales in N^ et ab N 
perpendicularis erigatur NS rectae EZ aequalis^ cir- 
cumque ^M et punctum tS segmentum circuli de- 
scribatur ASM] erit igitur semicirculo minus, quo- 
niam NS radio minor est. itaque angulus ad S 
punctum positus, qui rectis AS, SM comprehenditur, 
angulo ad E posito, qui rectis FEy E^ comprelien- 
ditur, aequalis erit. ponatui "pxa^Xj^x^^ L A^t^O =^Ti^^ 



214 OPTICORUM RECENSIO THEONIS. 

N aiprjQTJ^d^co fj NII fcfri tfj EZ^ xccl xsQLysyQdfpd^o jrepl 
tiiv AM xal tb 11 t^fjiia tcvxXov tb AUM' i6taL Sij 
fj jtQbg t(p n ycDvCa fj 7C£QiB%oiiLivri iTcb t&v AIIM 
t6ri tfj TCQbg tp E ycavLa^ JtsQLaxo^itnj Se i)7cb t&v 

b AEB. iTCal ovv ikd66cov fj TC^bg tp S 'cfjg ngbg t& O, 
t6ri S\ rj ^lv TCQog t& O tf] TCQbg trc3 E^ TcaQiaxo^ivy 
Sa x)7cb t&v ®E^ EH^ ij Sl TC^bg tp S tfj nqbg t& E^ 
TCaQLaxoiiivT] Sh vTcb t&v FE^^ ikd66c3v aQa ipavT^6ataL 
^l rj tf\g HS. TcdkLv ijcal ikd66c3V fj TCQbg ta E^ 

TCaQLaxo^ivrj S^ vTcb tcbv &EH tfjg JCQbg tm E^ ^a^L- 
axofiivrjg Sa 'bnb t&v AEB^ ikd66cov aqa (pavT^6ataL 
xal fj H& tfjg AB, 

Tcjv aQ^dtcjv OL tQoxol 6ta ^lv xvx^oaLSatg^ 6r^ 

6 S^ 7caQa67ca6^ivoL ipavofJvtaL. 

a6t(o yccQ tQOxig'^ oh SLd^atQOL al AZ^ BF. oix- 
ovv orav iiav ri aicb tov bfifiatog alg tb xivtQOV 

^ v£vov6a TCQbg d^d-ag ^ rc3 ijCLTciSc) -jj l'6ri tf] ix tov 
xivtQov^ l'6aL al SLa^atQOL (pavovvtaL^ cjg iv t(p TCQb 

avxov %^aG)QifiiiatL axaSaCx^ri' &6ta 6 tQOxbg 6 tof) ccq- 
^atog xvxlo£iSr]g (paCvataL toiitcov i)7caQx6vtc3v. jcaQa- 
g^aQO^iivov Sa tov aQ^atog xal tfjg aicb tov a^i^atog 
vavov6rjg aCg tb xivtQov dxxlvog ^i/jta TCQbg dQd^dg ov6rig 
t(p tov XQOxov iTCLTciSci) ^7]ta l'6rig tfj ix tov xivxQOv 

16 aixov dvL60L aC SLd^iatQOL g^avovvxaL bfioCcog SlA tb 
TCQb avtov SaLx^iv &6xa 7caQa67Ca6fiivog av (paCvoLXO 
6 xQoxog. 

3. 7] (pr.)] supra scr. m. 2 Y. 7] (alt.)] addidi; om. Vpv. 

8. (pavT^GStaC] -vrjastat in ras. m. 1 V. 11. AEE] AEB 

pv {A deformatum est in V). 13. Xf'] ^ia' Vv, ft^' p. 15. 

ftapsaTtaiisvov V. 16. dtafta-] in ras. m. 1 V. 18. ?]] corr. 

ex €l m. 1 Y. rg] corr. ex xm Y . 



OPTICORUM RECENSIO THEONIS. 



215 



et sumatur NO = EZ, circumque AM et O punctum 
describatur segmentum AOM, itaque iAOM= &EH, 
praeterea ponatur L ANU = AZE^ et sumatur 
NII = EZy circumque AM et 11 describatur segmen- 
tum circuli AIIM. erit igitur L AnM= AEB. iam 
quoniam L S < O et L = ®EH, L S =TEJ, 
minor adparebit F^ quam H®. rursus quoniam 
L &EH <,AEB, minor adparebit H® quam AB. 

37. 
Rotae curruum modo circulares modo oblongae 
adparebunt. 

sit enim rota, cuius diametri sint ^Zy BF. ita- 
que ubi recta ab oculo ad centrum ducta ad planum 

perpendicularis est uel radio 
aequalis ^ diametri aequales 
adparebunt, ut in proposi- 
tione praecedenti^) demon- 
stratum est. quare cum haec 
ita sunt, rota currus circu- 
laris uidetur. sed curru 
praeteruecto ubi radius ab 
oculo ad centrum cadens ne- 
que iam ad planum rotae 
perpendicularis est neque radio eius aequalis, diametri 
inaequales adparebunt rursus propter propositionem 
ante demonstratam [prop. 36]. ergo rota oblonga 
adparebit. 




1) H. e. per propp. 34—36. itaque forfcasse propp. 34, 86, 86 
in unam coniungendae erant. ^ 



216 OPTICORUM RECENSIO THEONIS. 

^Eav ^aysd^dg tc Ttgbg dgd^ag ^ tp i)7tox8L^ava) im- 
7ci8(p ii6tia)Qov^ tsd^fj d^ tb S^/ta iTtv tv ^rj^atov tov 
iitLTciSov^ xal ^ad^t^tfitatr tb bQa^iiavov iTtl xvxXov TtaQi- 

6 ^aQaCag^ t6ov aal tb b^io^avov d(pd^6ataL. 
"" ictco 6Q6^av6v ti ^iyad^og tb AB ^iatacoQ^taQov 
tov iytmidov^ Sfifia Sh S6t(o tb P, xal iita^avx^cj ij 
FB^ xal xivtQfo t& F^ Siaetr^iiLatv S\ tS FB xvxXog 
yayQccipd^io 6 B^. kiyco^ ort, iav iTcl tijg tov xvxkov 

10 TtaQifpaQaCag ^ad^i^tijtaL tb AB^ aitb tov F &^iiatog 
t6ov aal 6q)%"iq6atai, iital yccQ ii AB ictiv dQd^ii xal 
TCOial TCQbg f^v BF dQd^rjv ycovCav^ 7ta6ai aQa aC aitb 
tov xivtQOv tov r TtQbg tb AB ^ayad^og 7tQo6JtCjttov6av 
akkifikaig t6ag y<ovCag Ttoiov^cv. C6ov aQa tb 6Q(6^avov 

15 6(p%"^6atai, b^oCcog Sa xav aitb tov F xivtQOv ^iataco- 
Qog ai^ add^ata^ xal ix' avtrlg tb ofifio; tad^fj iTtl 
naQalkrikov ov t(p ^Qco^ivco iiayid^au^ xal iiataxiVY^tai 
tb ^iyad^og^ t6ov aal tb 6Q(o^avov (paCvatai. 

k%^\ 
20 ^Eav S\ tb 6Q(Ofiavov TCQbg oQd-ccg ^ rc5 vTtoxaifiivG) 
iTtLitiScp^ liad^cetfjtaL Sl tb o^^a inl xvxlov jcaQi- 
(paQaCag^ t6ov aal tb 6Q(o^avov (pavi^6atai. 

a6t(o 6Q(o^avov fi^v tb AB ^aticogov bv xal TCQbg 



1. Xr]'] ^B' YYj ny' p. 3. to] rc5 v. tov — 4. tcsql-'] 
dimid. eras. V. 4. Post nsQKps^siccg add. yisvtQov ^%ovtog rd 
biL^ux. p. 6. jistsoQmtSQOv V, ^istsaQov p; iistscoQoatSQOv V, sed 
corr. 8. 'nsvtQco] comp. Vv. 10. nsQLcpSQslag] comp. Vv. 
12. trjv] om. v. 13. tisvtQov] in ras. m. rec. V. 16. Ante 
&X^ii ^^^' 2 litt. V. iTti] supra scr. m. 1 p. 17. ^sysd^ v, 
sed corr. fistansi^vijtoci V, sed corr. ; fistanLvsltaL v, et p, 

sed corr. 19. Xd"'] ^y' Vv, ^id' p. 21. iitLTtsSo}] om. v. 



OPTICORUM RECENSIO THEONIS. 217 

38. 

Si magnitudo ad planum subiacens perpendicularis 

sublimis erecta est, et oculus in aliquo puncto plani 

ponitur, magnitudo autem, quae cemitur, secundum 

ambitum circuK mouetur, magnitudo, quae cernitur, 

semper aequalis cernetur. 

cematur magnitudo aKqua j4B plano sublimior, 

oculus autem sit F, ducaturque F-B, et centro F, 

radio autem FB circulus 

describatur B^. dico, ai j4B 

per ambitum circuli mouea- 

tur, semper eam aequalem 

a r oculo cemi. nam quo- 

niam A B perpendicularis 

est et ad BF angulum rec- 

tum efficit, omnes rectae, 

quae a F centro ad inagni- 

tudinem u4B adcidunt, angu- 

los inter se aequales efficiunt. ergo quod cernitur, 

aequale cemetur. similiter etiam si a F centro recta 

sublimis erigitur, et in ea oculus ponitur ad magni- 

tudinem, quae cemitur, positione parallela collocatus, 

et magnitudo mouetur, quod uidetur, semper aequale 

adparet. 

39. 

Sin quod cemitur, ad planum subiacens perpendi- 
culare est, oculusque per ambitum circuli mouetur, 
quod cemitur, semper aequale adparebit. 

cernatur ^B sublime positum et ad planum sub- 

21. Post icBQnpBQBiag add. yiivtQov ^jjjovrog t6 arntsiov^ %a^* 
o 6viL§dXXsL rb iLiysd^og rc5 iTtiiciSaa "^. 




218 OPnCORUM RECENSIO THEONIS. 

d^d^as ^Qog ro iycoxsL^svov invTCeSov^ b^^a d^ B6t(o 
ro r*, xal TtivxQG) ^lv rp B^ Sva6tifiiux,ti Se t& BF 
xiixkog ysyQag^d^co 6 F^. kiy(o^ Srt, iicv tb F [led'- 
i6tYitai iTtl xvxkov 7teQV(peQe(ag^ t6ov ael to AB 
. 5 (pavifi^etai, tovto S% (pave^dv i^tuv' na6ai yicQ al 
aicb tov r erjfiecov JtQbg tb AB nQo6%Cntov6aL axttveg 
TtQbg l6ag ycjVLag 7CQo67tC7Ctov6iv^ iTCeLSi/JTCeQ 17 TCQbg 
tp B dQd^TJ i6tLv. t6ov aQa tb bQco^evov (pavtl^etaL. 

r 
/*• 

10 ^Eav S\ tb bQafievov ^eyed^og /xi^ TCQbg dQd^i^g ^ ta 
vTtoxeLfievG) iTtLTteScD^ ^ed^L^tfjtaL Sh inl xvxkov TCeQi- 
(peQeCag^ avL6ov ael 6(p%"ifi6etaL, 

i6t(a xiixkog 6 A®^ xal eikr^^p^^G) ixl tfjg TceQc- 
(peQeCag avtov ^rj^etov tb A^ xal &ve6tdt(xi ^i^ n^bg 

15 oQd^ccg rp xvxXco ei^eta fi AZ^ o^/xcc Sl i6tc3 tb E. 
ksycOj otL fj AZ^ iav inl tijg tov xvxkov 7CeQL(peQeCag 
lied^L^trjtaL^ TCore ^eC^cov (pavrj6etaL^ 7Cot% ika66G}v, 

^tOL Sii ii ^Z iieCt,(QV i6tl trig ix tov xsvtQOv rj 
t6ri ^ ikd66cov. i'6tco TCQdtSQOv ^sC^cov^ xal ijx%(0 Sl& 

20 rov E xsvtQOv tfj AZ TCaQdkkrilog fj EF^ xal s6t(o 
t6ri tfj AZ i\ ET^ xal ijxd^co ajcb tov F 6rjfisCov inl 
ro i)7CoxsC^svov iTcCicsSov xdd^stog ij FH xal 6v^- 
^akkitco tp iTCLTCeSG) xatd ro H 6ri^stov^ xal iyCL- 
^sv^d^st^a ri EH ix^s^kifi6%^(o xal ^vfi^akkstco ty 

26 7CSQL(psQsCa xatd ro A^ xal '^x^^ ^^^ ''^^^ ^ ''^U 



5. tovtco V. ictL p. 6. t6] corr. ex rc5 m. rec. V. 9. 
11'] fte' p, fid' Vv. 11. di] ds tb (rw v) dQ^^isvov vp. 12. 
Post 6(p&T^astaL add. tioctcc itaqdXXT\kov &s6iv tjj i^ ^QXVS ^^stoc- 
§uZvov mg. m. 2 v. 14. gthisLov v. 17. Post nots (pr.) 

del. fiiv p. fisl^ov v. 18. ^rot dr}] ^ ds e corr. v, ijtoL 



OPTICORUM RECENSIO THEONIS. 



219 




quoniam L -B rectus est. 
adparebit. 



iacens perpendiculare, oculus 
autem sit F, et centro jB, 
radio autem BF circulus de- 
scribatur F^, dico, si F 
per ambitum circuli mouea- 
tur, AB semper aequale ad- 
parere. et hoc manifestum 
est; onmes enim radii a F 
puncto ad ^-B adcidentes 
angulos aequales efficiunt, 
ergo quod cemitur, aequale 

40. 

Sin magnitudo, quae cernitur, ad planum subiacens 
perpendicularis non est, per circuli autem ambitum 
mouetur, semper inaequalis cemetur. 

sit circulus AS, et in ambitu eius sumatur 
punctum ^, et ad circulum non perpendicularis eri- 
gatur recta ^Z, oculus autem sit E. dico, ^Z, si 
per ambitum circuli moueatur, modo maiorem modo 
miuorem adparere. 

aut igitur ^Z radio maior est aut aequalis aut 
minor. primum sit maior, et per E centrum rectae 
^Z parallela ducatur EF, et sit EF — ^Zy ducatur- 
que a puncto JT ad planum subiacens perpendicularis 
rHy quae cum plano in H concurrat, et ducta EH 
producatur concurratque cum ambitu in A, per A 



ds Vp. i}] del. punctis v. iist^ov v, sed corr. iarl Vp, 
ij] add. m. 2 V. 19. ficffoi; v, sed corr. 20. JZ] m. 2 v. 
21. tjj] m. 2 V. 22. ijtini$ov V, corr. m. 1, aS. i^ta^ 

tsvx^r)(5cc V, sed corr. 



220 OPTICORUM RECENSIO THEONIS. 

EF itaQttlXriXoq ii AB^ xccl i6t(o ii AB t^ jdZ t6rj. 
Xiy(o^ ZtL ii AB Ttae&v t&v iycl tfig tov xvxkov ^bqi- 
(psQSiag iisd^i^taiiiviov svd^SL&v ika60(ov (pavifi6stav, 
i7cs^svx^(06av yccQ aC TZ, EZ, BT^ EB. i%oiLSv 8\ 
^ ^ iv t^ TtaQaxstiiivc) rc3 As' d^scoQT^fiatc^ 8ti na6cbv tcbv 
Slc: tov E erj^siov dyofiivcov sid^st&v xal tcolov^cjv 
TtQog trj EF ycjvCav ika%i6tri i6tlv i^ '^^^ FEA. 
ijtsl ovv f^ FE tfi AB TtaQdkkrjkdg i6tiv, aXkcc xal 
t6rj^ xal ^i EA a^a trj FB [6rj ts xal naQaklrik6g i6tiv' 

10 7taQakkrjl6yQa^fiov aQa i6tl to BE, (Jtct: td: a%)ta Sij 
xal tb ZE 7taQakkrjk6yQa^fi6v ictiv, xal insl dst 
SsL^aLj 5ti ika66ov (paCvstav tb AB tov AZ^ SfjkoVy 
8tL 7tQ6tSQ0V Sst SsL^aL^ StL ^i i)7tb BEA yc^vCa ild6- 
6G3V i6tl tfjg iitb ZEA ycovCag, insl oi)v SiSsLxtaL^ 

15 StL 7ta6&v tcbv Sia tov E ^rj^sCov SLayo^ivcov sid^SL&v 
xal 7toLOv6&v TtQbg tfj FE ycovCag ila%C6tri i6tlv 'fj 
V7tb FEA^ ild66(ov aQa i6tl xal tfjg V7tb FEA fj 
V7tb FEA, ixxsC^d^co tS tov xvxlov fj^LxvxkCc) t6ov 
tb KAA^ xal sllritpd^G) Q^%)tofj tb xivtQov tb N^ xal 

20 xsC^d^co trj 'l)7tb FEA t6rj ycjvCa fj 'b^tb KNM^ ty Sl 
iTtb FEA t6ri fj V7tb KNO^ xal xsC6%^co trj AZ sxa- 
tiQa tmv OiV, MN l'6rj^ xal Slcc ^lv tov M trj KN 
l'6rj xal 7taQdkkrjlog ij^d^co fj MII^ xal i^ts^sv^d^co fj 
IIK' 7taQakkrjk6yQafiiiov ccQa i6tl tb NII xal t6ov 



3. -g fis-'] in ras. V. 4. Ss] ^rf v, 7. TCQog] supra 

scr. p. yaviag p. 8. SiXXd — 9. iativ] om. v. 9. icri p. 

10. i6tLv Vv. 11. iati p. dsZ] in ras. V, corr. ex di} 

m. 2 V. 12. otC] om. v, ag comp. m. 2. iXd66cov V, corr. 
m. rec. 13. SsZ] corr. ex drj m. 2 v. ^Xa66ov v. 14. 

i6tlv Vv. 17. ^Xa66ov v. i6tiv Vv. 18. tc3] corr. ex to 
m. 2 V. 19. rd (pr.)] corr. ex tc5 m. 2 v. to (tert.)] t& v. 

22. fisp] del. m. 2 v. 24. ictiv Vv. 



OPTICOEUM RECENSIO THE0NI8. 



221 



auiem rectae £1^ parallela ducatui jiB, sitqne j4B = A7.. 
dico, AB omnibua rectis, quae per ambitum circoli 
moueajitur, minorem adparere. ducantur euim I^Z, 
EZ, BT, EB. co- 
guouimus autem iu 
propositioiie theo- 
remati SXXVImo 
adnexa [p. 204, 
11 sq,], omnium 
rectarum per E 
ductarum et cum 
EF angulum effi- 
cieutium minimum 
angulum effieere 
AE, 9c. L TEA. 
iam quoniam FE 
rectaev)?Bparallela 
est, uerum etiam aequalia, etiam EA rectae FB et 
aequalia est et parallela; BE igitur parallelogrammum 
est. eadem de causa igitur etiam ZE parallelogrammum 
est. et quouiam demonstrandum est, AB minus ad- 
parere quam ^Z, manifestum est prius demonstrandum, 
esae /. BEA < ZE^. iam quoniam demonstrauimus, om- 
nium rectarum, quae per E punctum ducantur et cum 
FE anguloa efficiant, minimum angulum efficere EA, 
sc. L rEA, est L TEA < FEA. ponatur KAA aemi- 
circulo circuli aequale, et samatur centrum eius N, 
ponaturque LKNM=^ FEA, KNO = FEJ, et pona- 
tur ON = MN = ^Z, per M autem rectae KN 
aequalia et parallela ducatur MII, et ducatur ilff; 
Nn igitur parallelogrammum est et '5a':e.\.\!&V'0'©»ssssMa 




222 



OPTICORUM RECENSIO THEONIS. 




xal 5(ioiov rp BE. naXiv Sia rov O rfj KN t^rj Tcal 

TcaQciUrjkog %^C3 fj OP^ xal iTCstevxd^co ^ PK' ro PN 

aQa TCaQakkrjXd- 

yQafiiwv t6ov ra 
5 %al ofioidv i6rL 

rc5 ZE. xal ine- 

^svxd^co^av aC Sia- 

ydfviOL al PN^ 

IIN. &6rB xal rj 
10 vTcb KNIl yfovCa 

rijg 'l)7cb KNP 

ycovCag iM66G)v 

i6rCv. xaC ianv 

il (ihv {)7cb KNn t^rj rfi vnb AEB^ fj Sh vnb KNP 
15 t^ri rfi i)7cb JEZ' ikd66(ov HcQa fj 'bnb AEB rfjg 

i)7cb jdEZ. &6re xal rb AB fiayed^og roi) z/Z fieye- 

d^ovg ika66ov 6(pd^6eraL. 

b^oCcog di) SeC^ofiev^ ort i^ "^^ '^VS ^^ iXd66(ov 
i6rl rfjg ZA t6rjg re xal iXd66ovog rfjg ix rov xev- 

20 rQov v7CaQ%ov6rjg. 

akXd Sij i6r(o rj jdZ rfj ix rov xevrQCw i!6rj^ xal 
xars^xevd^d^G) 7cdvxa rd avra: rotg 7CQ6reQOv^ xal xeC^d^cj 
ra roi) xvxkov rj^ixvxkCa) i'6ov 'fj^ixvxhov ro SKA^ 
xal el?,7j(pd'G} avrov ro xevrQOv ro N. xal i^cel ij AZ 

25 t6rj V7c6xeirai rfj ix rov xevrQOv rov xvxkov^ t6rj ccQa 
i6rlv rj AZ rfj ®N. xal xeC6%^(o rfj fiev V7cb FEA 
yc3vCa t6rj rj i^cb SNK^ xal ^xd^cj rfj SN 7caQdkXrjkog 

5. iexiv Vv. 7. diayayviat p. 11. KNP] corr. ex KN 
m. 2 V. 13. iatlv] ietl p. 14. ij 8s — 15. AEB] mg. 

m. 2 V {iisiiisvov). 15. l'ari] ^^- ^- i^-daacov aQo] &ats 

%ai V. AEB] AEB iXdaaav iati v. 17. ildaaav V, sed 



OPTICORUM RECENSIO THEONIS. 



223 



BE aequale et simile. rursus per O rectae KN 
aequalis et parallela ducatur OP, ducaturque PK'^ 
PN igitur parallelogrammum est parallelogrammo ZE 
aequale et simile. et ducantur diagonales PN, UN 
itaque L KNH < KNP, est autem L KNH =- AEB, 
KNP -= ^EZ, quare L ^EB < JEZ, ergo etiam 
magnitudo AB cemetur minor magnitudine z/Z [def. 4]. 

iam similiter demonstrabimus, esse BA < Z^, ubi 
Z^ radio uel aequalis est uel etiam minor. 

iam uero ^Z 
radio aequalis sit, 
eademque omnia 
comparentur^quae 
antea, et ponatur 
semicirculus SKA 
semicirculo circuli 
aequaliS; centrumque 
eius sumatur N. et 
quoniam supposuimus, 
AZ radio circuli ae- 
qualem esse, erit 
AZ= ®N 
ponatur igitur 

LSNK=rEA, 
ducaturque KS rectae 
SN parallela, suma- 
turque KS == SN, et ducatur S^, ponatur autem 

corr. 18. JsXaaeov v, sed corr. 19. iGxiv Vv, mg. dgj-O*?}- 

GBtai m. 2 V. 21. ftf' Vv. 23. rd] rc5 v. 25. r«] corr. 

ex tfjs V. 26. V7t6\ 'bitb t6 Vvp. rEA] e corr. V. 27. 
fari] i- in ras. V. 




224 OPTICORUM RECENSIO THEONIS. 

rj S®j tfj dh {)7tb t&v FE^ t^rj xeLdd^G) ij iTcb r&v 
SN^^ Tcal t^ ®N TCaQdkkrjkoQ i^xd^co ^ ^O, xal t^rj 
trj ®N &(prjQrl6d'a) 'fi ^O^ xal i%Bl8vx^^ fi O®' JtaQ- 
5 aXXrjldyQafiiiov &Qa i6tlv ixdtsQOv t&v 0^, ®K^ xaC 
i6tiv t6a te xal ofioia toig EZ^ EB. &6ts xal fj ^lv 
ijtb SNJ yovCa t6rj i6tl tfj 'bnb FE^^ ii 81 iTtb 
@NK t6rj i6tl ty {)7tb FEA. ikd66(ov 8\ ij iitb FEA 
trjg inb FEA' ikd66(ov aQa xal fj {)7tb SNK trjg 

10 iTtb ®NA. [xal] i7ts^svx^(o6av aC 8iay6vL0L ai SN^ 
ON' ikd66(ov aQa xal fi vTtb SNS trjg iitb ®NO. 
t6rj 81 rj (ilv i)7tb ®NS r^ i^b AEB^ fj 8s iitb SNO 
tfj 'bitb /:1EZ' ikd66(ov aQa xal fi 'bnb AEB trjg 
'bitb AEZ. ika66ov aQa h(p%"Yi6staL tb AB fiiysd^og 

16 tov AZ fisysd^ovg' oTtSQ sSsl 8st^aL. 

akXd 8ii l6t(o 'ij ^Z iXd66(ov tfjg ix tov xsvtQOv 
tov xvxAov, xal xats^xsvd^d^cj td a^vtd totg TtQdtSQOv^ 
xal xsL^d^cj ta tov xvxXov fj^LXvxkLG) t6ov tb SM^ 
xa\ slkifi^pd^ci tb XBvtQOv tov xiJxXov tb iV, xal d^prjQi]-' 

20 ad-co dTtb tfjg ®N tfi AZ t6rj rj NS^ xal xsC^d^co tfj 
ILBV V7tb FEA ycovCcc t6rj fj {jTtb SNK^ tfj 8^ vTtb 
FEA t6rj rj 'bitb &NA^ xal l6tcj t6rj sxatSQa tcbv 
NK^ NA tfj AZ^ xal Vjx^(o 8Ld ^isv tov K tfj NS 
t6rj xal jtaQdkkrjXog 'rj KO^ xal iTtB^s^vx^co 'fj OS^ 8Ld 

25 8\ tov A trj SN naQdXXrjXog 'fj AII^ xal iTts^Bvx^cj 
rj n^' 7taQakkrjX6yQa[i[iov ccQa i6tlv ixdtSQOv tcbv 
KS^ SA., xaC i6tL ro iisv K3 t^ EB t6ov ts xal 

3. @NJ] mut. in ©NA m. rec. V, ©iV p et add. z/ m. 2 v. 

@N] corr. ex @ m. rec. V. JO] AO Y. 4. JO' 

AO Y. 5. 9d] ^0 p, @A Y. 6. totg] tjj p. i^ ^fV 

om. V. 7. @N^] @NA V. iativ Vv. 8. iativ Vv. 

10. 0NJ] @NA V. yLcci] om. Vv. 14. ildaaoDv p. 



OPTICORUM RECENSIO THEONIS. 



225 



i SN^ = FEjd, ducaturque z/O rectae ®N parallela, 
sumaturque z/O = ®Nj et ducatur 00; itaque utrum- 
que ®jdy @K parallelogrammum est et parallelogrammis 
E Z, EB aequalia et similia. quare etiam L SN^ == FE^y 
0NK=rEA, uerum L TEA < FEJ. itaque etiam 
L ®NK< ®N^, ducantur diagonales ^N, ON ita- 
que etiam L ^NS < SNO. uerum L ^NS = AEB, 
0NO = ^EZ. itaque etiam L AEB < JEZ, ergo 
magnitudo AB minor magnitudine AZ cemetur; quod 
erat demonstrandum. 

iam uero zfZ radio circuli minor sit, eademque 
comparentur, quae antea, et semicirculo circuli aequale 
ponatur ©M, sumaturque centrum circuli Nj et a 
GN auferatur NS rectae AZ aequalis, ponaturque 





Sf 



L &NK=rEA, L SNA = rEA, et sit NK=NA=AZ, 
ducaturque per K rectae NS aequalis et parallela KO, 
et ducatur OS, per A autem rectae tSN parallela AII, 
et ducatur il^*, utrumque KS, SA igitur parallelo- 



16. fi-g' Vv, fi.J' p. rfjg] corr. ex rfjL m. 2 V. 18. t6] 
rw V. 25. rjl corr. ex rf)g V, rfjg pv. 26. Post ri ras. 

llitt. V. 27. iarLv Vv. 

Bnolldes, edd. Heiberg ©t Mengft. '^H. "^"^ 



226 OPnCORUM recensio theonis. 

SfiOLOVj tb Sh SA r& EZ' &6re xal ycovia 'fj imh 
SNK tdri rfi imh FEA^ ij 8e 'bnh &NA ry 'bnh FEJ. 
(lei^cov Sh ii inh FE^ rrjg iTch FEA' iieCt,(ov &Qa 
Tcal ii 'bjch 0NA rijg inh &NK. iTtete^dxd^atfav aC 
5 NOj Nn- xal ii i)nh SNO HcQa rflg ijch SNH iXa6- 
6(ov i6rCv. t6ri Sh 'fj ^ev 'bTch SNO rfj ifTch AEB^ 
'fj Sh 'bnh SNII rfi 'bnh ^EZ' ikd66(ov aga xal i^ 

"^ 'bTch AEB rfig 'bnh AEZ. xal §Jiinerac 'bith ^iv rfjg 
AEB rh AB [liyed^og^ 'bich S% rfjg ''Imh ^EZ rh AZ. 

10 ika66ov &Qa 6(p%"ifi6 erai rh AB fiiyed^og. rov AZ ^e- 
yid^ovg' oiceQ iSec Set^av. 

(ia\ 
"EfSn rvg rdjcog^ o5 rov Sfi^arog ^ivovrog^ rofj S^b 
hQCjfiivov [led^L^ra^ivoVy t6ov ael rh 6Q(0(ievov (paCverau 

15 e6r(o y&Q 6Q(b(ievov fihv ro BF^ b(i(ia Sh rh Z, d(p' 
ov 7CQo67CV7crir(o6av dxrtveg aC ZF^ ZB^ xal TceQi- 
eiki^^pd^^o rh ZBF rQCycjvov xvxAco rc3 ABZ. Xiy(o^ 
ort ro -B r [led^i^rdiievov iicl rfjg rov yQa(pivrog x^vxXov 
7ceQi(peQeCag t6ov del 6Qad"ij6eraL. (leraxeC^d^cj yaQ rh 

20 BF iicl rov FA^ xal iTCe^evxd^ca 'fj AZ. oixovv t6ri 
i6rlv 'fj Br 7ceQL(piQeia rfi FA 7ceQL(peQeCcc, t6ri aQa 
xal fi P ycjvCa ry U ycovC(x. rd Sl 'bnh t6(ov ycovLcbv 
6Q(h(ieva t6a (paCveraL. t6ov aQa (paCveraL rhBF r(p J!^. 

2. vTt6 (sec.)] 'b' in ras. m. 1 V. 3. iLtftfcoi; (utrumque)] 
liBltov V. VEA] triv FA v (inter T et ^ ras. 1 litt.). 9. 
VTcb JEZ] ^EZ p. 12. iia'] iiri' p; ^f V et v m. 1; ils' 

V m. 2. 13. iisvcovtog v, sed corr. 15. Post Z eras. d V. 
17. ZBT] BZr p. 18. inl] i- in extr. lin. v. 21. tfj] 
tfjg V. jtSQLtpsQSioc] -a add. m. rec. V. 22. Post ij eras. 
7} V. rj] tfjg p. yoavia] yoavlccg p. taov aQU apaivstcci 

tk 8\ vTtb iaav yoavi&v bQanLSvoc taa (pccivstai v, corr. m. 2 lit- 
teris apy adpositis. 'bTci] vnb t&v p. 23. to] tm v. FJ] 
JT supra scr. m. 1 V. 



OPnCORUM RECENSIO THEONIS. 



227 



grammuni est, ei KS parallelogrammo EB et aequale 
est et simile, tSA autem parallelogrammo EZ] quare 
etiam L ®NK = FEA, L ®NA ^ FE^. uerum 
L TEA > FEA. itaque etiam L ^NA > ®NK. 
dueantur NO, NII. itaque etiam L SNO < SNII. 
uerum LSNO = AEB, LtSNn=AEZ] quare etiam 
L AEB < jAEZ. et ab angulo AEB magnitudo AB 
cemitur, a AEZ autem AZ, ergo magnitudo AB 
minor adparet magnitudine z/Z; quod erat demon- 
strandum. 

41. 
Locus est, unde oculo manente, mota autem magni- 
tudine, quae cernitur, haec semper aequalis adparet.^) 

cematur enim BF, 

oculus autem sit Z, a 

quo radii adcidant ZF, 

ZB, triangulusque ZBF 

circulo ABZ compre- 

hendatur. dico, si jBr* 

per ambitum circuli de- 

scripti moueatur, sem- 

per eam aequalem cemi. 

transponatur enim BF 

ad FAj et ducatur ^dZ. 

itaque arcus BF arcui 

r.d aequalis est. quare etiam L P= ^- q^ae autem 

ab angulis aequalibus cemuntur, aequalia adparent 

[def. 4]. ergo BF magnitudini Fz/ aequalis adparet. 




1) In figura litteras P, Z permutauit v, pro -27 in Vp 
eet O. 



228 



OPTICOEUM EEOENSIO THEONIS. 



"E6tL rig xdTCog^ oi tov o[iiiatog fied^Ldtaiievovj tov 

Sh 6Q(0(iivov (iBvovtog^ del t6ov tb 6Qd)fi6vov (paCvatai. 
idtfo yaQ 6qg)^£vov ^iv 
6 tb BF^ ofi^a dl tb Z, &(p^ 

ov 7CQ067CV7Cth(o6av axtlveg 

aCZB^ Zr^ xal TCBQLBiXri^pd^c^ 

toBZr tQiycovov tfii^fiatL 

xvxkov t& B Z Fj xal 
10 ^BtaxBL^d^cj tb Z o(i[ia iicl 

tov A^ xal iiBta7CL7Ctitc3- 

6av aC axttvsg av z/5, 

jdr. ovxovv l6ri fj P ycovLa tfl 2J' iv y&Q t& ait^ 

tinjfiatL b16l. ta 8\ vTcb t6c3v ycovLcjv 6Qcl)(iBva t6a 
15 (paCvBtaL. l'6ov ccQa tb BF SLa 7Cavtbg (paCvBtaL tov 

oii[iatog [iBd^L^taiiivov i7cl tfig BFjd 7CBQL(pBQBCag. 




^y. 

"E6tL tLg to^cog^ ov tov b[i^atog [iBd^L^taiiivov^ tov 
Si bQcoiiivov ^ivovtog^ avL6ov tb ^QcofiBvov (pavBltaL. 

20 ietG) yccQ 6Q(OfiBvov tb Kjd^ Bid^Bta Sh ij BF ^vfi- 
7cC7Ctov6a tri Kd 7CQo6BxPakko^iv7]^ xal BiXi^^pd^G} tfjg 
/IF xal tijg FK ^i^rj avtkXoyov fj FZ^ xal iTca- 
i^BvxQ^G} 7] ZK xal fj Zz/, 7CbqI Sb tijv K^d tfiri^a 
yByQcccpd^G) d^Btav i%ov tijv ycjvCav i^pccipBtaL di^ 

26 rijff BF Bid^BCag^ i7CBC7CBQ i6tCv^ Sg fj ^F 7CQbg ti^v 
FZ^ ovtG)g fj ZF 7CQbg tf^v FK xbC^^^g) ovv tb 8(ifia 
i7cl tov B 6riiiBCov^ xal TCQo^^B^kri^Q^G^^av at z/-B, BK^ 



1. fi-^'] ft-O"' Vp, V m. 1; iLto' V m. 2. 2. -Q'i6rtt-'\ in 

rae. V. 11. tov] mut. in rd m. rec. V. {LBtanntrixtoGcci V, 



OPTICORUM RECENSIO THEONIS. 



229 



42. 

Locus est, unde oculo moto, magmtudine autem, 
quae cernitur, manente haec semper aequalis adparet. 

cernatur enim BF, oculus autem sit Z, a quo 
radii adcidant ZB^ ZF, triangulusque BZF segmento 
circuli BZr comprehendatur, et oculus Z ad z/ trans- 
ponatur, radiique rursus adcidant /JB, ^F. est igitur 
L P= 2]'^ nam in eodem segmento sunt. quae autem 
ab angulis aequalibus cemuntur, aequalia adparent 
[def. 4J. ergo BF semper aequalis adparet, si oculus 
in arcu Br.d mouetur. 

43. 

Locus est, unde oculo moto, magnitudo autem, 
quae cemitur, manente haec inaequalis adparebit. 

cematur enim K^^ 
recta autem sit BF 
cum K^ producta 
concurrens, et me- 
dia inter ^F, FK 
proportionalis suma- 
tur rZj ducanturque 
ZK, Zz/, et circum 
K^ segmentum de- 
scribatur angulum 
acutum comprehendens; continget igitur rectam BF, 
quoniam est z/F: FZ = ZF: FK, iam oculus in B 




corr. m. rec. 12. al (pr.)]om. p. 13. P] post ras. 1 litt. V. 
14. slat.] supra -at, ras. V . 17. fty'] v' Vp, v m. 1; fi-J' v 
m. 2. 20. To] rc5 v. 24. d^stccv] in ras. V. ^x^v v., 

-ov in ras. V. 27. TtQoasyL^s^lriG^iii^^v ^. 



230 OPTICORUM RECENSIO THEONIS. 

i7ce^£vx^(o Sh 71 U^. (ydxovv t6ri 'fj ymvCa tf} 2J 
ycavCa' iv yi^Q x& ait^ tii^^iuctC el^iv, xaC i6tLV fi 2J 
t^g B ycavCag fisC^cov xal 'fj ccQa ycovCa trjg B 
^eC^cov i6tCv. tov aga oii(iatoQ inX tov Z ovtog fist^ov 
6 (pavattaL tb K^d i^jtSQ iTcl rot) jB. 

iiS\ 

Tb d' aitb (iv(ip7]6stai^ xav xaQdXXrjkog fj fi yQa^ifiij 
t& bQcoiiivcD (isyid^SL^ itp^ -^g tb b^ifia (isd^C^tatai. 
§6tG) yaQ TCaQcckXrjXog fj BF tp 6Qco(iivG) rra ^Z, 

10 xal Si%a tstyiiffi^ci i^ z/Z xatk tb K^ XQbg d^d^ag Sh 
avTjxd^cj 'fj KN. %eC6%'Gi ovv tb 6(i(ia inl tov iV, Tial 
iTCa^s^vx^co^av at N^^ NZ^ tcsqI Sh tijv ^Z t(ifi(ia 
yeyQa^pd^c}^ 8 Si^atav tijv O^ A yovCav. inel ovv 
SiccfiatQog i6tiv ij KN^ xal TCQbg dQd^dcg a%^ axQag 

15 '^xtai fj KN tfi BF^ ij BF ccQa itpdittstai tov ANZ 
t(i7j(iatog. (istaxsC^d^G) Sij tb b(i(ia inl tov P, xal 
TCQO^Ps^lij^d^cj^av aC rZ^ F/l , iTCstsvx^o Sh rj PZ. 
o^xovi/ l'6rj rj 0^ A yovCa tfj P yoivCcc. rj Sl P tfjg U 
ycovCag (isC^cjv i6tCv' (isC^cov &Qa xal rj ^,^ A tfjg U. 

20 tdc Sh 'bicb (isC^ovog yavCag 6Qc!)(isva (isC^ova (paCvstav 
(ist^ov ccQa (pavsttai tb AZ tov '6(i(iatog iicl tov N 
xsL(iivov ^TCSQ ijcl tov r. tov aQa (i(i(iatog iicl trjg 
Br (isd^L^ta^iivov jcaQaXXrjkov ov6rjg tfj AZ ccvl6ov 
(paCvstai tb bQG3(isvov. 



2. bIgi p. 3. Ante B ras. 1 litt. V. aQoc\ in ras. V. 
4. iazi V' 5. iTti'\ supra scr. m. 1 V. B] e corr. V. 6. 



fid'] va Vp, V m. 1; ^t}' v m. 2. 7. rf] supra scr. V. 10 

di iviJ^t-O"©] diccvoix^a) v. 13. A] postea ins. V. 15. KN] 

Jt e corr, m. rec. V. 16. tov] mut. in to m. rec. V. 



OPTICORUM RECENSIO THEONIS. 



231 



puncto collocetur, et adcidant JByBK, ducatur autem 
2J. itaque /. = 27; nam in eodem segmento sunt. 
et /, 27 > B ; quare etiam /. > jB. ergo KJ maius 
adparebit oculo in Z posito quam in B. 

44. 

Idem autem eueniet etiam, ubi recta, per quam 

oculus mouetur, magnitudini, quae cemitur, parallela est. 

sit enim BF magnitudini, quae cemitur, ^Z par- 

allela, et in K recta ^Z in duas partes aequales 

secetur, perpendicularis autem erigatur KN, oculus 

igitur in N collocetur, 
ducanturque N^, NZ, 
circum ^Z autem seg- 

mentum describatur, 
quod angulum O -{- A 
capiat. iam quoniam XiV 
diametrus est, et ad KN 
perpendicularis in ter- 
mino erecta est BFj segmentum JNZ contingit BF, 
iam oculus ad F transponatur, et adcidant JTZ, F^y 
ducaturque PZ. itaque /. +^ = -P- uerum L P> 27; 
quare etiam /. -f- -<4 > 27. quae autem ab angulo 
maiore cemuntur, maiora adparent [def. 4]; quare ^Z 
maius adparebit oculo in JV" posito quam in F. ergo 
si oculus per B F magnitudini ^Z parallelam mouetur, 
quod cemitur, inaequale adparet. 




17. jCQO^s^Xijc&oaaav Vpv. 
lisiiava v. 



19. icti p. 20. fisL^ovcc] 



232 



OFIICORUM RECENSIO THEONIS. 



''E6tL tLg tdTtog xoivdg^ iv c5 rc!: i!6a fisysd'rj avL6a 
(paivstai. 

l6ta) yccQ C6rj 'fj BF tri FJ^ xal tcsqI [ihv ti^v BF 
6 rjfiixvxXiov ysyQa^pd^c} tb BZFj tcsqI Sh f^v Fjd t(irliia 
^st^ov 'f}(iixvxXLOv^ xal snst,sv%%'a)6av au ZB^ ZFj Z/1, 
ovxovv i\ iv tS i^iiLXvxXCG) ycovLa (isl^oov i6tl tfjg iv 
ta fiSL^ovL tfirlfiatL. t& Ss vtco ^SL^ovog yiovLag 6Q(h- 
lisva [iSL^ova tpaCvstaL' (iSL^cav aQa i] BF tr^g F^ 
10 (paCvstaL' ^v Sh xal H^rj, i6tLV aQa tdnog xoLvdg^ iv 
o5 tcL i'6a (isyid^rj avL6a (paCvstaL, 



''E6tL tLg tditog xoLvdg^ a(p^ o^ rcJ: avL6a fisyid^rj t6a 

(paCvstaL, 
15 ' s6t(o yaQ (isC^(x)v 'li BF trjg Fz/, xal tcsqI (ihv f^v 

BF (ist^ov iiiiLLXvxXCov tfirjfia ysyQd^pd^cj^ tcsqI Ss 

tijv Pz/ ofiOLOv ta 

tcsqI tijv BF^ tovt- 

i6tL Ssxofisvov yc3- 
20 vCav l'6rjv tfj iv t& 

BZr^ i7Cs^svx^(o6av 

Ss at ZB, Zr, ZA, 

ovxovv iicsl l6aL 

sl6\v ai iv tolg bfioCoLg t[i7]^a6L ycjvCaL ccXXifikaLg^ t6aL 
25 sl6l xal at iv totg BZF^ FZ^ t^T^iia6L ycovCaL dkXr]- 

kaig, td Sh vtco l'6cov ycovL&v 6Q(b^sva i'6a (paCvstaL' 




1. fi-f'] v^' V, V m. 1; /x-O-' V m. 2; vy' p. 6. |LtftJcoi; v. 

7. iGtiv V. 8. {LeilovC\ iisiSoavL v, sed corr. 9. ^Bi^av} 

lisltov V. 12. fi-g'] v8' p; vy' V et v m. 1 ; v' v m. 2. 13. 
foaj supra scr. m. rec. V. 15. iisttov v. 




OPTICORUM RECENSIO THEONIS. 233 

45. 

Locus est commiinis; ubi magnitudines aequales 
inaequales adparent. 

sit enim Br^Fzfj et circum BF semicirculus 
describatur BZr^ circum Fz/ autem segmentum semi- 

circulo maiuS; du- 
canturque ZjB, ZJT, 
Zz/. itaque angulus 
in semicirculo posi- 
tus angulo in seg- 
mento maiore posito 
maior est. quae 
autem ab angulo maiore cemuntur, maiora adparent 
[def. 4]. itaque BF maior adparet quam F^; eadem 
autem aequalis erat. ergo locus est communis, ubi 
magnitudines aequales inaequales adparent. 

46. 

Locus est communiS; unde magnitudines inaequales 
aequales adparent. 

sit enim BF > F^, et circum BF segmentum 
describatur semicirculo maius, circum F^ autem seg- 
mentum illi simile, h. e. quod angulum angulo m BZF 
posito aequalem capiat, ducanturque ZBy ZF, ZA, 
quoniam igitur anguli in segmentis similibus positi 
inter se aequales sunt, etiam anguli in segmentis 
BZT, TZA positi inter se aequales sunt. quae autem ab 
angulis aequalibus cemuntur, aequalia adparent [def. 4]. 
oculo igitur in Z puncto posito BT magnitudini TA 
aequalis adparebit; eadem autem maift^ ^'?^\.. ^s^ 



234 



OPnCORUM RECENSIO THEONIS. 



tov &Qa H^^atog tLd^s^ivov iitl tov Z 6Yi[isiov t6ri av 
^alvoLto fi BF tri F^d' i6ti 8% (ist^cov. idti tcg &Qa 
tdicog xoLvdgj atp^ oi tcc avL6a ^eyi%"ri t(fa ^aCvatai. 

5 Ei6C tiveg tdjtov^ iv olg tk avi6a (leyidifj dtJo elg 

tavtb ^vvted^ivta t6a ixatiQO) t&v icvC^cov (paCvatai. 
i6to yd^Q [isC^cov fj BF tflg F^^ Tcal tcsqI tccg BF^ 

r^ 'fjfiLXiixha ysyQdtpd^co^av xal tcsqI Slrjv tijv B^. 

o^xovi/ t6ri fi iv 
10 trc3 BA/:1 iiiLLXV' 

xXCg} ycovCa tfj iv 

t^ BKF' dQd"^ 

yaQ idtiv sxatiQa 

ait&v, t6ri &Qa 
15 (paCvstaL ii B F 

tfi B^' d)6avtcog 

Ss xal ^i B/1 tfj r^ t&v 6^(idtcov iTcl t&v BAJ^ BKF^ 

TTjA 'fiiiLxvxkCcov xsifiivcov. sCdC tivsg &Qa tdicoi^ iv 

olg ta &vL(Sa fisyid^rj Siio sCg tai)to Cvvtsd^ivta t6a 
20 sxatiQC) tmv avCCcjv (paCvstaL, 




[irj. 

EvQstv tdTCovg^ acp^ hv tb t6ov (liysd^og i](iL6v (pa- 

vsltaL t) titaQtov fiiQog xal xad^dkov iv r« Sod^ivtv 

I6ya}j iv c5 xal ij ycovCa tifivstai, 

25 i6to y&Q sifd^sia fj AZ^ xal tcsqI tijv AZ ys- 

yQdg^d^cD t(ifl(ia tv^dv^ xal iyysyQd^pQ^G) sig aitb ycavCa 



2. tpcLivBixo V, corr. m. 1. tvg\ in ras. m. 1 V. 4. ftf'] 
vb' p; v8* V, m. 1 V; va m. 2 v. 6. avvti^^ivta p. 7. yLBl^ov v. 



OPTICORUM RECENSIO THEONIS. 235 

locus est communis, unde magnitudines inaequales 
aequales adparent. 

47. 

Loca sunt^ ubi magnitudines inaequales duae con- 
iunctae utriuis magnitudinum inaequalium aequales 
adparent. 

sit enim Br> F^, et circum Br, F^ semicirculi 
describantur, item circum totam jBz/. itaque angulus 
in semicirculo BA^ positus angulo in semicirculo BKF 
posito aequalis est; nam uterque rectus est. itaque 
BF magnitudini B^ et rursus jBz/ magnitudini FJ 
aequalis adparet oculis in semicirculis BAJ, BKF^ 
rZ^ positis. ergo loca sunt, ubi magnitudines in- 
aequales duae coniunctae utriuis magnitudinum in- 
aequalium aequales adparent. 

48. 
Loca inuenire, unde magnitudines aequales dimidiae 
adpareant uel quarta pars uel omnino secundum da- 
tam rationem, secundum quam angulus secatur. 





recta enim sit AZ, et circum AZ segmentum 
quoduis describatur, in eoque angulus K inscribatur, 

8. yay^iaqp-O-G) p. 11. rj] rt}v v. 12. BKF^ post B 

ras. 1 litt. V. 16. BT] F e corr. V. 21. fi,7] ] vs' p; 

vb' V, m. 1 v; v^' m. 2 v. 22. ^v\ o-u v. 



236 OPTICORUM RECENSIO THEONIS. 

'fl K^ rfj 8h AZ tffrj ifStio ii BF^ xal TtBQl xiiv BF 
TteQLysyQdfpd^co tii7}^aj b da^stav f^v rijg K ycovtag 
illLi^Biav. oi)KOVv rj K ycovca 8i%ka6ia B0tl trlg A 
ycavLag. SiTtka^Ca aQa tpaCvBtai 'fi AZ tfjg BF td)v 
6 6/Ltfc.c^trQ}i/ iycl t&v AKZ^ BjdF %BQi(pBQBiG)v xblijlbvcov. 



Tcbv 160) tdxBL (pBQOiiivcDv xal btcI tfjg aitrjg Bvd^aCag 

bvtcjv 7CQo6L6vtcov ^BV TCQog tb S^^a ro tBkBvtalov 

jCQorjyBWd^aL So^bl^ TcaQakXai^dvtcov 8\ tb ^hv tcqo- 

10 rjyoiiiiBvov iTcaxokovd^Btv j ro dh ijcaxokovd^ovv tcqo- 

rjysWd^aL S6^bl. 

tpBQB^^^cD y&Q L^otax&g td BF^ ^dZ^ KA^ xal aicb 
tov M S^^atog 7CQ067CL7CtBtca6av dxttvBg at MF^ MZ^ 
MA. ovxovv ^BtBCjQotdtrj i6tl xal dB^LCotsQa tav aicb 
15 rov (i^(iatog dxtCvcov 7CQ067CL7Ctov6G)v ij MF' ro aQa 
BF d6^BL ^CQorjyBL^d^aL. 7CaQakka^dvtcov dh tcbv BF^ 
AZ^ KA xal B7cl t&v N^^ 11 P^ ET ysvo^Bvcov 7Cqo6- 
7CL7CtBtG)6av dxttvBg at MN^ MII^ MU. oixovv 7Ca6cbv 

tCbV d7Cb tOV a^^atOg dxtCvCJV 7CQ067CL7CtOV6G)V dB^LCO- 

20 tiQa i6tlv fj ME^ dQL6tBQd 81 fictUov fj MN' &6tB 
xal ro ^lv ZT 7CQoriyBt6^aL 86i,BL^ i^caxoXovd^Btv 81 
tb NS' ro ^lv aQa BF ^CQorjyoii^Bvov i7cl rov NS 
yBv6}iBvov 861^BL i^caxokovd^Btv^ ro 8b AK i^caxokovd^ovv 
i7cl rov 2JT yBv6^Bvov 86^bl ^CQorjyBt^d^at. 



3. iatlv V. 6. f^^"'] vj' P; vs' V, m. 1 v; vy' m. rec. v. 

8. tsXsvtsov V. 13. M] supra scr. m. 1 V. 14. ^stso- 

QODtdtri V, corr. m. rec. ; ^stsoQOtdtri v. 23. do^si] mg. 

m. 1 y. 



OPTICORUM RECENSIO THEONIS. 



237 



sit autem BF = AZj et circum BF segmentum de- 
scribatur, quod partem dimidiam anguli K capiat. ita- 
que i K = 2^. ergo AZ duplo maior adparebit 
quam BF oculis in arcubus AKZy BjdF positis. 



49. 

Magnitudinibus aequali celeritate motis et in eadem 
recta positis ad oculum adcedentibus ultima prae- 
cedere uidebitur, praetergressis autem praecedens sequi, 
sequens praecedere uidebitur. 

aequali enim celeritate moueantur BF, ^Z^ KA^ 
et ab M oculo adcidant radii MT^ MZj MA. MF 

igitur e radiis ab 



>^ 




oculo adcidentibus 
maxime sublimis est 
et ad partes dextras 
positus; quare BF 
praecedere uidebi- 
tur. praetergressis 
autem BFy^Z^KA 
ad N^, nP, UT 
radii MN, MH, MU 
adcidant. ex omni- 
bus igitur radiis, qui ab oculo adcidunt, maxime ad 
partes dextras positus est MUj ad sinistras autem MN] 
quare UT praecedere uidebitur, NS autem sequi. 
ergo BF magnitudo praecedens, cum ad NS per- 
uenerit, sequi uidebitur, AK uero sequens, cum ad 27 T 
peruenerit, praecedere uidebitur. 



238 OPTICORUM RECENSIO THEONIS. 



r 

V . 



^Edv XlV(OV (pBQO^iviOV 7tXBl6v(OV &vCfS(p xdxBL 6VIJL- 

TCaQccfpBQrjtaL btcI rd airoi xal tb (ififia^ td: fihv rp 

'6fLfLaxi l6oxa%S)Q (pBQ^iiBva 86i,BL B^xdvav^ x& 8\ ^Qa- 
5 SvxBQOV Blg xoi^vavxCov (piQB^d^ai^ xa S\ d^axxov Blg 

xa jtQorjyojifiBva, 

(pBQi^d^ca y&Q dvC6p xd%Bi xd 5, P, ^, 

Kal ^QaSvxaxa filv (pBQi^fd^cj xb Bj xb 

Sh r i6oxax&g xp K SfifiaxL^ xb Sb A 
10 %axxov xov T^ d%b S\ xov K Sfifiaxog 

7tQ067CL7txBX(X)6av dxxtvBg aC KB^ KFj 

K^. ovocovv xov '6fifiaxog 6vfi7taQa- 

(pBQOfiivov xotg 5, r, ^ xb fihv F xaxd 

xijv FK dBl (pBQ^fLBVov B6xdvaL S6i,BL^ xb S\ B iTCo- 
15 kBL7c6(iBvov Big xoi>vavxCov S6^BL (pBQB^d-aLj xb Sb jd^ 

ItcbX d^axxov xov F tpiQBxaL^ S6i,BL Big xovfi^tQo^d^BV 

7tXBtov ydQ dTtb xov F d7to6XYi6BxaL. 




va\ 



'Edv XLV(DV (pBQOfiBvcjv S La(paCvrjxaC xl fi'^ (pBQ^fiB- 
20 vov^ S6^BL xb fiij (pBQ6fLBvov Blg xoivavxCov (psQB^d^aL. 

(pBQB6d^co ydQ xd B^ A^ fLBVBXC^ Sh xb P, xal d7tb 
xov Z ofifLaxog 7tQo67tL7tXBXco6av dxxtvBg aC ZB^ ZJT, 
Z^. ov^covv ro (iBv B (pBQ^fLBvov lyyLOv l6xaL xov JT, 
ro Sb A d^to^coQovv 7tOQQ(oxBQOv, &6XB S6^BL xb r 

25 Blg XOVVavxCoV (pBQB6d^aL. 



1. v'] V7]' p; v^* Y, m. 1 v; vd' m. 2 v. 2. evnTtccQa- 

fpBQUt ai V, corr. m. 1. 3. r6] corr. ex rc5 V. to5] x6 v. 

4. (pBQ6yLBvoi V, sed corr. 6. cpaiQsad^ciL v. 9. leaytccx&S 

^ V, sed corr. m. 1. 11. KB] BK seq^. lac. 1 litt. v. 14. rX] 



OPTICORUM BECENSIO THEONIS. 239 

60. 

Si compluribiis magnitudinibus inaequali celeritate 
motis in pari;es easdem etiam oculus mouetur, quae 
eadem celeritate mouentur, qua oculus, stare uide- 
buntur, quae minore, in partes contrarias moueri, quae 
maiore, praecedere. 

moueantur enim inaequaU celeritate B, Fj A^ et B 
minima celeritate moueatur, F eadem, qua oculus K^ " 
/1 maiore quam P, ab oculo autem K radii adcidant 
KBj KF, Kjd, itaque si oculus in partes easdem 
mouetur, in quas B, JT, A, magnitudo F, quae ad FK 
semper mouetur, stare uidebitur, B autem, quae re- 
manet, in partes contrarias moueri uidebitur, ^ uero, 
quoniam celerius mouetur quam T, praecedere; magis 
enim a F remouebitur. 

51. 

Si motis magnitudinibus aliquot interlucet aliquid 
non motum, hoc in partes contrarias 
moueri uidebitur. 

moueantur enim B, A^ maneat autem 
F, et a Z oculo radii adcidant ZJ5, 
2jTj Z^. itaque B magnitudo cum 
mouetur, magnitudini T adpropinqua- 
bit, A autem, quae recedit, longias 
distabit. ergo T in partes contrarias moueri uidebitur. 

T seq. lac. 1 litt. v. 16. iithl^ inl v. %axxxov v. 18. va^ 
v^' p; vr{ V, m. 1 v; vz' m. 2 v. 19. ^t}] in ras. m. 1 V, 
om. p. 28. ^yystov V. 24. &7tox(OQOvv] &7toxa}Qsixa} V. 

25. sig] om. p. 





240 OPTICORUM RECEN8I0 THEONIS. 

* 

Tov H^iicctog iyyuov rov 6q<d^8vov 7tQ06i6vrog 86^bi 
xo 6q(6ii€vov rji^fi^d^ai, 

dQcc^d^co y&Q rb BF tov 6(i(icctog iTtl tov Z xsLfiBvov 
6 vTcb rcbv ZB^ ZF &xtLV(bVj 
xal ^staxsL^d^c^ tb o^(ia iyyiov 
tov BF xal l6tco btcI roi) ^d^ 
ocal bQd^d-o) tb aitb vnb t&v 
jdB^ jd r axtLvc^v. (yixovv 
10 ^sL^cov ri ^ yc3VLa trjg Z yo)- 
vCag, t& 8% vTcb fi€L^6vcov yo)- 

vLcbv bQtbfisva (iSL^ova cpaLvstaL' 86^sl &Qa rjv^ficd^aL 
tb BF tov S^^atog iicl rot) ^ '6vtog ^tcsq inl tov Z. 

vy\ 
15 Tg)v 1'6g) tdxBL cpSQO^evcov ra 7c6qqc3 8oxbl ^Qa- 

SvtBQOV <pBQB6d^aL, 

(fBQB^d^ca yaQ l6otaxcbg ta 5, K &g inl ta Z (liQrj^ 

xal aitb roi) A 6^^atog dxttvsg ilx^co6av au AF^ AA^ 

AZ, ovxovv ro K iXdd^ovag ijiSL tdg d%b tov A 

20 o^iiatog dxttvag 'fiyfiBvag iJTCBQ ro B' sXattov ccQa 

,. 8Ld6trj^a 8LBkBij0BtaL xal 7CQ6tSQOv icaQakkd66ov tijv 

AZ (illJLV 86^BL taxitBQOV (pBQS^d^aL, 

v8\ 
Tov o^^atog TCaQa^pSQO^BVov td 7c6qqc[) tG)v bQco- 

25 ^BVC3V XataXBLTCB^d^aL 86i,BL, 



1. vjS'] I' p; vQ'' V, m. 1 v; vz' m. 2 v. 2. ^ysiov V. 

3. riv^Bla^ocL V, sed corr. 6. iyyBiov V. 9. Ante ^F 

ras. 2 litt. v. 10. iistSov v. 11. iisitmvcov V, sed corr. 



OPTICORUM RECENSIO THEONIS. 



241 



52. 

Magnitudo, quae cernitur, oculo ei adpropinquante 
aucta esse uidebitur. 

oculo enim in Z posito cernatur BF sl radiis ZB, 
ZFj et oculus magnitudinem BF propius transponatur 
sitque in ^, et eadem magnitudo a radiis ^B^ ^F 
cernatur. itaque L ^ > Z, quae autem ab angulis 
maioribus cernuntur, maiora adparent. ergo BF oculo 
in ^ posito maior esse uidebitur quam in Z. 

53. 

Magnitudinum aequali celeritate motarum remotio- 

res tardius moueri uidentur. 

moueantur enim aequali celeri- 
tate By K ad partes Z, et ab A oculo 
radii ducantur ^F, A^, AZ. ita- 
que K radios ab A oculo ductos 
minores habebit quam B. ergo 
distantiam minorem permeabit et, 
cum uisum AZ prius transgrediatur, 
celerius moueri uidebitur. 

54. 

Ubi oculus praetermouetur, res, quae remotiores 
cernuntur, remanere uidebuntur. 




13. ovxoi\ corr. ex ^yb\mroq V. 14. vy'] |a'p; |'V, m. 1 v; 
vi' m. 2 V. 18. ^r] seq. ras. 1 litt. V, corr. ex ^BP t. 

21. xat — 22. qpf^fiff-O-ai] om. V. 21. TCaqaXXadGfav vp. 

23. i;^'] IjJ' p, lu' V, vri in ras. m, % ^. 

Enclides, edd. Heiberg et Menge. "^TL. "^^ 



242 



OPTICOBUM RECEN8I0 THEONIS. 



l(ft(o yaQ &^^a tb B^ &g)' ox) fix%G}(!av &xrtv€g at 
BF^ Bjd^ BZ^ bQfb^Bva 8\ xcc 
K^ A, oixovv rov Hfifiatog 
7CaQag)SQOfjLevov TCQbg rotg F 
6 fi^QSiSL d^atrov TtaQsks^d^ovrac al 
fS^sig rb K VpcsQ rb A, 86%si 
aqa rb K vuoXsiits^^ai^ rb 8\ 
A slg r^yhvavrCov (psQS^d^ai^ 
tovrs^riv hg snl r& jCQbg r& 
10 Z iiiQYi* 




vs\ 



Tcc av^avdfisva rcbv fisysd^&v Eyyiov 8oHst r& Sfi- 
liari 7CQ06dys6d'aL. 

l6r(o y&Q 6qco(isvov ro FB vTcb r&v KBj KF 
15 axrivcov^ ocal rjv^rl^d^ca rb BF rp B^d^ xal aicb rov K 
b^^arog 7CQ067CL7crsrc3 axrlg ii K^, oixovv ^sl^cjv 'fj 
i^cb ^KF ycavLa rfjg {)7cb BKF ycoviag. rd^ 8h t&jto 
lisCtfivog ycovLag bQm^sva syyLOV (paCvsraL, syyLOv &Qa 
86^SL slvaL rb F^ iJ7CSQ ro BF. 



20 vg\ 

TO^a ^il iv rc5 avrffll a7C06r7]^arL xstraL ^'^ 7taQ- 

dkkrjka xsC^sva rcbv axQC3v /lm^ xardkkrjka xsLfisvcov r&v 

^i^cjv ^rjSh i7c' svd^sCag 8vr coi/, ro oAov cfXW^ ^"^^ 

^sv xotkovj 6r^ 8h xvQrbv 7C0Lst. 

26 bQd^d^ca ydQ ra B^ Fj ^ rov ^(ifiarog i7cl rov K 



6. pLSQsaLv Vv. 7. ro df] corr. ex tov ds V. 11. vs'] 
ly' p, 1/5' V, vd'' in ras. m. 2 v. 12. ^yysLOv V. 14. TB] 
Br p. 16. rii^sle&ia v, sed corr. 16. iistSov v. 18. 

iyytov (pr.)] lyysiov V, ftetfora p, om. v. (paivstai\ om. v. 



OPTICOBUM EECENSIO THEONIS. 243 

oculus enim sit J5, a quo ducantur radii BFy B^d, 
BZ, cernantur autem K, A. itaque ubi oculus ad 
partes F praetermouetur, uisus magnitudinem K prius 
transgredientur quam A, ergo K remanere uidebitur, 
A autem in partes contrarias moueri, L e. ad partes 
ad Z positas. 

55. 

Magnitudines auctae oculo adpropinquare uidentur. 

FB enim a radiis KB, 
KF cematur, et ^JT magni- 
tudine B^ augeatur, et ab 
oculo K adcidat radius K^. 
itaque L^Kr>BKr. quae 
autem ab angulo maiore 
cemuntur, propiora uidentur. ergo F^ propius esse 
uidebitur quam BF, 

56. 

Quae nec parallela sunt nec in eadem distantia 
posita extremis nec mediis respondentibus nec in 
eadem recta positis, totam figuram tum concauam tum 
conuexam efficiunt.^) 

cernantur enim B, r^ ^ oculo in K posito, radii- 




1) Cum Graeca sensu careant, Latina in lioc quoque uestigia 
eorum sequi coguntur. 



%yyiov (alt.)] ^yyeiov V. Ante ^yyiov (alt.) add. tic $1 iibI- 
iovoc kavt&v oidpLSva to^ 6(i(uctog inocv^fkvsa^ai do%oii6i' wxl 
tic ai)^ccv6iiBva ccga t&v fisys&mv dd^SL 7tQ0(tciysa&ai t& 6(iiuctt p. 

20. vg'] gd' p, ly' Vv (y del. m. 2 v). 2&, ^i\^ ^j 



244 



OPTICOBUM RECEKSIO THBOHIS. 



xtifitvmf, xal xqo09i9tixaaav eacrtvts tU KB^ fj', 
KjJ. (fhxovv rb olov ej(^fi(ta xotlLov av d^^ttv slvai. 
lurtaavaie&a di{ xdXiv rb b^ajievov xal fyyiov xc£e9a> 
Tov 3/ifucTo?. oixovv To ,dBV 96^Bi xvqiTov eivai. 



vf. 



''1 



TeTQayavov vXKpxovTog i&v Sixb r^g 6wa<p^g zCtv 
Sia^ixQoyv Jtpog 6Q&«g xig ava%^'^ t^ toO xexpayavov 
ixiaiSm, ixl dh xaikTjS xt9^ xb Spfia, ai te aXtvQal 
xov XBTffayavov xal at StdftiTffoi Caat ipavovvxat. 

10 Itsxa yag Texffdyeivov xb fZ, d 

XttX didittXQOi i^-i&BiSav aC PZ, 
KiJ, xal &xb xov & XQog bQ&ag 
^X^a tA ixixiSip ij SS, t6 Si 
tififta xs^fffrw ^«1 xov B, xal jrpoff- 

15 xiXxixBieav dxxXvtg at KB, BJ, 
Br,BZ. oixovv Svo aC Z&, &B 
dvo Tats r&, &B teai sieiv. tlel 
ds xal at yaviat at ztQiBx^jievaL 
vn ait&v teai, xovxiexiv at «pog 

20 t^ ®- teri dga xttl i^ ZB ^deig 

T^Brpdaei.. di& xd ttind dij xttl " " 

fl KB x^ B^ terj ieriv. Svo S^ at ZB, BF Svel 
r«rg KB, ^B tSai tialv ixoTdQa ixaxiqa' xai sCetv 
at SidfittQoi taaf aexe xal at XQbg tra B ymviat teai 

25 ieovxai. xd di 'bx.b teav yaviav bga^tva tOa ipaivsTttf 
teai aQa tpavovvrat aX ts StdfitTQOi xccl at nltvQal xov 
XBXQaydtvov. 




2. &v] BCripBi; 
iS' V, m. 1 y; 1«' E 
ai seg. lac, 8 litt. t 



. Vvp. 3, lyyttov V. 5. v^'] ^f' p; 

3 V. 8. ial Se] ijitl *ij y. tairiis] 

9. fooi p. 15. BiJ] B e corr \. 



OPTICORUM RECEN810 THEONIS. 



245 



que adcidant KBj KFy K^, itaque tota figura con- 
caua uidebitur. iam rursus magnitudo, quae cemitur, 

K 





transponatur oculoque adpropinquet. ergo ^BF con- 
uexa uidebitur esse. 

57. 

Dato quadrato si in puncto sectionis diametrorum 
recta ad planum quadrati perpendicularis erigitur, in 
eaque oculus coUocatur, et latera quadrati et dia- 
metri aequales adparebunt. 

sit enim FZ quadratum, ducanturque diametri 
rZj KJy et in ad planum perpendicularis erigatur 
SB, oculus autem in B ponatur, adcidantque radii 
KB, B^, BT, BZ. itaque duae Z0, @B duabus 
r®y ®B aequales sunt. uerum etiam anguli ab iis 
comprehensi, h. e. qui ad ® positi sunt, aequales sunt. 
ergo etiam ZB=Br. eadem de causa etiam KB = B^. 
quare duae ZB, BF duabus KB, ^B singulae singuKs 
aequales sunt; et diametri simt aequales; quare etiam 
anguli ad B positi aequales erunt. quae autem ab 
angulis aequalibus cernuntur, aequalia adparent. ergo 
et diametri et latera quadrati aequalia adparebmit. 

16. &B — 17. r©] om. V. 17. 0JB] corr. ez BF 

22. KB] e corr. m. 1 v. iazl p. 23. z/B] v et m.;: 
corr. ex ^r V, -J0 p. 26. xf[ tat v. 




246 OPTICORUM RECENSIO THEONIS. 

. Tflg dh iath r&v dfifidrfov ixl r^ (fwccgyijv r&v 
8iaiLirQ(ov fLi^re n:Qbg dQd^itg oii67ig rS hct/xid^ fii^s 
tffrjg ixariQcc r&v &7cb rrjg 6wa^fig nQbg r&g yciyvCag 
rov rarQaydjvov &yo^ivc3v fiifcs t6ag ycaviag mQUxov^rjg 
5 ^sr^ avr&v aC dcdfisrQOL avv6ov cpavovvrav. biLoCcog 
yicQ daC^ofisv rcc ^viifiaCvovra^ xad^djcsQ xal iv rotg 
xvxXoig. 



2. /iifrf (pr.)] ft^ p. 4. tcag] corr. ex tariq m. rec. V. In 
fine: ra itQb djtTLx&v E^Tilsidov q)ll£ riXos stXri(ps sifdonovvzog, 
& d6^oc p. 



OPTICORUM RECENSIO THEONIS. 247 

Sin recta ab oculo ad punctum sectionis diametro- 
rum ducta neque ad planum perpendicularis est neque 
utrique rectae, quae a puncto sectionis ad angulos 
quadrati ducuntur^ aequalis neque cum iis angulos 
aequales comprehendit, diametri inaequales adparebunt. 
nam eodem modo, quo in circuKs, rei rationem de- 
monstrabimus. 



■'■•3 




-f 

f 



SCHOLIA 

IN 

OPTICOEUM RECENSIONEM 

THEONIS. 



f; 



SCHOLIA 

IN 

OPTICOEUM EECENSIONEM 

THEONIS. 



>JI 



i' 



i •■ 



Ad praefationem. 

1. TovtB6tt xat& 6vvi%eiav p. 148, 18 — 19] ov rovro 
iovas XiyBiv to xatcc 6wi%BLav ^yovv 6vvBX&g xccl 
ixo^ivog &bC' Bi^rj y&Q av ivavtCov t& iv dia^tifi^ati 
g^iQB^d^aL xal ix 8La6trjfidt(DV tavtag 'b%aQ%BLV' kiysL 6 
Sl xat& 6vvi%BLav th iq>Bi,Yig ^BtaTcCictBLv xal ^ij 7ts- 
Tckavrj^ivcog^ &XXcc Kat& (ABtdfia6Lv 7tQOl'ov6ag xal ^sd'- 
L6ta^ivag, 

2. "EfpsQBv altCag p. 148, 22] ^yovv altLd^ata Sg 
^ij xat& X6yov Xsy6^svov aitLfo^svog a{)t6, 10 

3. Olov ycovCaL p. 154, 2] x&vtsvd^sv S^a tb iv 
SLa6t7J^a6L tag Sfl^SLg (piQS^d^aL^ v6bl 8h tavta t& 
SLa6t7J^ata fiQax^^tata S6ov ol6v ti i6tL (idXL^ta^ S6ov 

talg TCQog rw Sftftart y^ovCaLg iyyC^SL X0QQ(htSQ0v 

tov '6^^atog &sl iisCtjco yCvstaL .... xivtQOv y&Q rov 16 
6fi^atog vocrvfiivov &vdyxrj tag ^fl^SLg xcovosLd&g g)iQS- 
^d^aL xal 7CQOtov6ag ^aXXov dXXijXcov ^xC^s^d^aL^ 8 xal 
drjXov ahLOv yCvs^d^aL tov nav fiiysd-og ixsLv tL SLd- 
6tri(Aa^ &fp oh oix ^QataL, i^ixQL ^lv y&Q iyyLOv Sv 
^st^ov fi tov t&v 6iIjbg)v SLa6trl(Aatog^ bQcctaL^ ijcsLS&v 80 

1. v^ 2. y\ 3. v^ 



14. Ante noQQArsQov septem litterae, quas extricare ne* 
). 15. 6tiaTog v^ Ante xi 
&vdiy7irig?). 16. icvay%ri\ comp. v^ 



queo. 15. ditaTog v^ Aute ksvtqov comp. inGertina (ML^ 



252 



SCHOLIA 



61 TtOQQ&rsQov y6v6(Aavov iist^ovL iavtov dva6ti/iiLati 
t&v Hipsc^v ivfdxjl^ iiSrj ^rjSa^&g airtov t&v ^sav 
ig)a7Cto^sv(x)v Siic th 7caQS[iLnsnt(oxsvaL tp Sta<ft7l(iati 
avtG)v o{)x (>QataL, 

5 Ad definitiones. 

4. Ta 'bnh iiSL^ovog ytovlag 6QC)(ASva fASL^ova q>aL- 
vstav oif^ savt&v^ aXXic ^sC^ova Srjkovdti^ rj si icjQato 
{)%h bi,siag ytovCag* olov cjg iv 'bTtoSsCy^atv i6tc^6av 
di5o tQCycjva t6a t& BF^^ BKA^ (asC^c^v Sh i6tG) ii 

10 tov BF^ tQiyfbvov TCQhg rc5 B ycovCa^ naQ^ o i^ tov 
BKA iCQhg ta a^dta ^rj^sCp, Xsya)^ ott th BF^d tQC- 
ycDvov 'bich ^sC^ovog ycjvCag hQCj^svov^ TcaQ^ o th KBA^ 
^st^ov q)aCvstav tov KBA Sid^ th tijv 'bnh FB^d yco- 
vCav slvau ^sC^ova tf^g 'VTch KBA. -i} th ^sC^ova iv- 

15 tavd^a th 6vyxQLtLxhv avtl hcXov xsttat wg slvat th 
^sC^ova (paCvs^d^aL avtl toi) ^sydla (paCvs^d^at^ &67Csq 
th ivavtCov ta 'bnh ild66ovog yovCag d^scoQO^v^sva 
^LXQo: (paCvstaL xal ta iith l6rjg l6a, 

5. MstS(x)QOvg ^ev a7tkG)g 
20 axttvag tdg ^axQdg dvo^d^SL xal 

'bil^riMg^ lAStsoQOtSQag S\ tovtcov 
avr^i/ TtdkLV tdg ^axQotSQag ts 
xal 'biprjkotSQag' olov mg iv i)7to- 
SsCyfAatL i6tco6av tQCa fAsysd^rj 
25 dXkifiXcDv ditsiovta Ixavhv S^d- ^J^ ^ 

6trj^a td BF^ ^Z, KA^ xal 7tQ067tt7ttstc36av iic^ 

4. V* (ad def. 4). 5. V* (ad def. 5). 




1. Post ysvo^isvov del. . . . t&v Sia6ri]y,ccTog ysvoiisvov v*. 
6. axdXiov V*. 7. SriXovdTi] supra scr. m. 1 V^ 15. 

cvyyQLTinov V^. 



m OPTICORUM RECENSIONEM THEONIS. 253 

avta SipSLg at BN^ ^N^ KN. Xeyco^ Zti C6ov 
^eysd^&v tovt(ov iTCoxei^evGJv xal aTch tov N 0rifi£LOVj 

XaO"' S i0tL tb Sflfia^ tcbV aKtivOV 7CQ067CL7CtOV6G}V 

yLEtecoQOtEQa i6tlv 'fj ^lv BN dxtlg tflg ^N^ ij Sh ^N 
trig KNj xal 6^0L(x)g av tovto iTcrjQxev^ sl xal sts^aL 6 
TcksCovg avt&v ^6av, 

6. Tovts6tLV otav to avto dLa 7cXsl6vg)v ycovL&v 
b^ataL' t6ts ya^ sx tcbv bips(ov axtlvsg aitalg iQSL- 
86fisvaL SLa tcXslovcdv av XsyoLvto Sqccv tb bQ(x)fisvov, 

Ad prop. I. 10 

7. ^SL yccQ tb bQ(6^svov a7c66ta6Lv tLva s%slv TCQbg 
tb Sftfta* ovtco yaQ xal oQad^T^^staL^ og^ sl' ys ^rjSs^Lav 
a7c66ta6Lv ^xsl^ ov% oQad^TJ^staL, 

Ad prop. n. 

8. Oi> ydcQ av sItcol^sv p. 156, 17] sl yicQ ilsv- 16 
6ovtaL SLa tcbv F^ ^, yCvstaL tQCyoovov S%ov Siio {)7C0- 
tSLV0ii6ag^ hv 'fj ixtbg v7CotsCvov6a ^sC^cov yCvstaL tr^g 
ivt6g^ v^cstsd^rj Sl l'6rj. 

9. Mij d^OQvfisCtco y&Q 'Ijnag tovto^ O7ccog tb fihv 
BFjd tQCy(ovov iTcl 7ckiov rjvscoxtaL xat& Tckdtog^ tb 20 
S\ BKA 6tsv(htSQ6v i6tL, 7CQ&tov ^lv ydQ tov 6toi' 
%SL(otov ^rjtovvtog t6a xal 7caQdXXriXa vostv td (pac- 
v6iLSva^ sCtcsq tb BKA tQCycovov xatd 7cdvta i(p7]Q[ioiB -^ 
rc3 BF^ tQLy(A)V(p^ ovx av ^6av di5o, dXX' d)g ^ 
i(paCvovtOy dX)^ oi)S\ 7caQdXXrjXa' vvv d' ovtcog^ iiq %l 

6. V* (ad def. 7). 7. M^Rqru(Ft). 8. V«q. 9. V". 



12. yuQ W] %cd ydq Ru, yccQ t. VI. iwii^^ ^KOtj/^i 



254 SCHOLIA 

B%SL^ tsd^BvtcDV 6vii^aCvBL f^v Ixd^B^LV ifpttQ^d^Biv ai>totg' 
xal yicQ TtaQdkkrjkd tB b16l ta tQtymva^ xal tb BKA 

tQfy(OVOV TtkBOVBXtBt t& [lilflXBL t&v BK^ BA yQa^^&v^ 

xaC i6tL dLct tavta t6ov to stBQ 

5 10. ^EnsLdifi^ 56aL av ixttvsg inl to F^ %qo6- 
7Cb6(o6lv^ i^(otBQaL l6ovtaL tov KA fti) 7CQo67cCntov6aL 
ax)tip* &6ts iTtb 7cksL6v(Dv &QataL tb Pz/. 

11. ^AH^ SrjkovdtL ^BXQL tmv K^ A TtSQdtcov ik- 
^•ov^aL 6tifi6ovtaL xal i(p' iavtdg avaxka6d"il6ovtaL .... 

10 6triQCt,ov6LV ^ &l)J &}g ^•..tL iTtsl iyy&csQ^v i6tL tb 
B T^ tQCyc^voVj xal TtXsCovsg btl^sLg tovtp 7tQ067tB6ovvtaL^ 
xal axokovd^c^g axQL^B6tSQ0v bQa%^if^6BtaL^ tovtB6tL ^aX- 
Xov ^ ro stSQOv bQa%^if^6BtaL. 

12. nkBL6v(ov oipscov p. 156, 23] si ds vTtb 7tXsL- 
15 6vG)v S^scav^ xal {)7tb 7tlsL6vc3V yavicbv, 

Ad prop. III. 

13. "l6G)g stTtOL ttg av^ hg^ STtSLSii ov ^6vaL aC 
BF^ B^ 7tQo67tC7ttov6LV axttvsg 7tQbg tb F^ {Lsysd^og^ 
akka xal aklaL 7tkBt6taL ^sta^i) tcbv F^ ^, ots aq)L6ta- 

20 ^svov tov Fjd ^sysd^ovg ov 7tC7ttov6Lv at BF^ B^ 
dxttvsg^ 7tQo67ts6ovvtaL aC fista^i} tov fis6ov 7CQ067ts60V' 
6aL axttvsg. ksyofisv ovv 7tQbg tbv ovtC3 a7C0Qif^6 avta^ 
8rfc, sl xal 7tQbg ^LXQbv a(pE6trjx6tog tov F^ ^syid^ovg ov 
7tQo6paXov6Lv at BF^ B^ axttvsg^ aXX' aC ^sta^v tov 

25 ^s6ov^ xal i7tl 7tkst6tov a(ps6trix6tog tov tOLOvtov ^sys- 
d^ovg ovd^ a[ ^sta^i) tov ^s6ov 7tQ067CS6ovvtaL Sid: tb 
7ckatvvB6^aL tb iiBta^v tcbv toiovtcov ^ipBcov Std6tri^a 

10. VM^FRqst (ad p. 156, 23). 11. V». 12. U\ 

13. R(MAFqrstu, Vat. m. 2). 



8—10 non intellego. 17. stnoCl Mqr, «tjriy RFrt. 



IN OPTICOEUM RECENSIONBM THEONIS. 255 

&^uSTtt(tdvov TotJ HEjidQovs ovtos &Qi0(i.ivov Xttvrbg 
fiEyd&ovg, 

14. T&v j-cp SiaeT7}(tiiT<av ij (laXXov AaoeTaaemv 
itQo%mQovOS}v iezat (isza^i) dideT7j(ta, oi aC (broffTtfflfig 
6ict ro ^' SilXijXeiV &sto0%i0^vai ohx a^ovtai.. 6 

Ad prop. IV. 

15. "Eetm TQfymvov d(f9-oymviov t6 KBZ dg&iiv 
i%ov T^r JE^og rp B,' toai iJi l9zm6av al RT, F^, ^Z, 
Xttl ^rfgftf^^Orotfav aC FKy ^K. (pri(il Stj, Szt ^ M 

Tijs N fuCtav ietiv, ^ di JV 10 
T^s S ffj;*'^ y^P ^^ ^"^ I^ 
Tji dK xagdlltj/.os i5 ^-^- 
i«Tiv a(ftt, its ^ ^r aQbg 
PB, ovtas ij KA n^bg 
tifv JB. fUTj Sh ^ Jr Tfj 18 
FB- tan a^a ml ij KA 
Ty AB. xal ijtsl ipfrrj iSTiv 
ij 3Fp6g T^ B, (iti^mv ij FA 
tijs AB, Toxrtiatt r^g AK- 
aOte xal ymvla f{ M (iti^mv so 
i«rl Tijs O. SilX& )5 O TiJij ietl tfj N' ivall&i, ydQ sttSiv 
Kal ^ N apa T^? M iX^Woyv iexlv. itdXiv &Jlb toH /i 
Tfl ZK %a^&lhfjkos ^%^m i^ JIl' fpavtqhv ffif, Stj ij P 
^i%mv ioTlv ip&^g. meze xdXtv 6[ioims Sei^ofisv, Srt 
^ 77^ ^fi^ioi' iffrl r^s /7ff ' raOTS xal yavia ^ JV 36 
U. E'. 16. Y(Vat.qt); ad p. 168, 20. 

1. 6Qii!iihov R. 7. ie#oj'«bi'MH'] i" T. 6q»<^v} ± V. 

19. ifls (pt.)] rfl V? 21. Ante ri)s ras. 4 litt. V. ivcdii£ V. 

23. iUJ e corr. m. rec. Y. iXaeatav] comp. corr. ex p^mp 

m. tec. V. 23. Ante P erafl. ij V. -^ 




256 SCHOLIA 

rflg 2J. aW 'fj 2 t^ S i6xtv t6ri' xal 'fj N &Qa r^g S 
^SL^cov i6xCv, 

16. ^'E6tc3 t6a dLa^tT^j^attt i%l ^Lccg eiQ^eiag tcc AB^ 
BF^ r^j xal avifi%%^G) tfj A/1 JtQog dQd^dcg fj AE^ icp^ 

5 ^g xsL^d^G) oftfta tb E, keyco^ StL ^st^ov (pav^^atac 
tb ^lv AB rov BF^ tb Si BF roi) FA, 7CQ06%L%tB- 
tG)6av y&Q dxttveg at EB^ EF^ EA^ xal ^^fO-cj Sci: 
tov B 6ri^6L0v tfj FE sid^SLcc TcaQakkrjkog '^ BZ dL& 
tb SsvtSQOv tov SKtov, koLTcbv i6taL t6ri 'fj AZ t^ 

10 ZE, ^SL^cov Sh 'fi BZ trjg ZA Sik tb ^ai^ova ycovtav 
i)7toteLVBLv' ^SL^cov ccQa xal trjg ZE, iieCtfjOV ccQa xal 
^ S yovCa t7]g K, aXk& trj K t6ri ii A Sl^c ro eivaL 
ivaXkd^' ^eC^cov aQa i^ ® ^^^ '^VS ^* fiet^ov ccQa 
6(pd"il6etaL tb AB tov BF, o^oCcjg Slcc tov F ai%eC- 

15 6rig 7caQakX7]Xov trj AE tfjg FH SeL%%^ri6etaL tb BF^ 
5tL ^et^ov q)av7J6etaL tov FA, 

17. Al^c tb f^v AF {)7toteLveLv xal tijv M ^eC^ova 
ov6av xal tfjg AK tfjg 'V7CoteLvov6rig ffjv O^ 'fj Se 
^eC^cov TtkevQct ffjv fieC^ova ycovCav 'bicoteCveL, 

20 'fi Se elg tccg icaQaXXifjXovg evd^eCag ifi7cC7Ctov6a tccg 
ivakXa^ ycavCag t6ag ccXXifiXaLg tcolbL 

Ad prop. VI. 

18. Kdd^etog &Qa i6tCv p. 162, 3—4] TC&g 'fj KM 
xdd^etog i6tLV iitl tijv MA^ SeC^o^ev oiitcog' i%el aicb 



16. v^ in mg. sup. (ad ipsam prop. 4 add. htBqa xovtov 
av(o icTCodsL^ig); est opt. uet. prop. IV. 17. q (ad schol. nr. 15 
p. 255, 20 et 21). 18. R, q fol. 109 (add. fijrft iv tc5 f' &£a- 
QT^^au) (M^Arsu, Vat. m. 2). 



24. Post iTtsi add. g?" (pvv) R. 



\ 



IN OPTICORUM RECENSIONEM THEONIS. 257 

tov K iTcl ro {}7CO}t£Liiavov iTCLTCsSov Kad^stog fjxtccL 'fj 
KA^ xal %Qhg 7Ccc6ccg ccQa t&g ccTCto^ivag aitijg sid^ecag 
xal ov6ag iv r« i)noK£i^iv(p iTCvjciSco ii KA dQd^ag 
7COL7^6sL ycovCag, iTCsl ohv iicl ti^v ZA xdd^stog ^xtac 
fj AM^ xal TCQog tijv AM i^ KA d^d^iiv Tcoiif^esi ycj- 5 
vCav, iTCs^svxd-o ccTcb tov A Tcal iicl th A fi AA' ;cal 
TCQhg ccQa tijv AA ij AK dQd^ijv tcov/i^sl ymvCav. insl 
ovv tQCyG)v6v i6tLv dQd^oyovLOv th KAA 6Qd"^v S%ov 
tijv {)7ch KAA ycjvCav^ th &Qa aich t7]g KA 'bico- 
tsivov6rig tijv 6Qd"^v yc^vCav l'6ov i6tl tolg ccTch t&v 10 
KA^ AA. TcdkLv iTCsl tQLya}v6v i6tLv dQd^oyovLOv th 
AMA dQd^ijv i%ov ti^v 'bich AMA yovCav^ th ccQa ccsch 
trig AA t6ov i6tl totg d^h t&v AM^ MA, th ccQa 
ocTch tijg KA [60V i6tl totg anh tov KA^ AM^ MA. 
aXXa totg ccich t&v KA^ AM t6ov i6tl th anh tr^g 16 
KM' tQCyovov yaQ i6tLV dQd^oyovLov th KAM dQd^ijv 
s%ov tiiv iynh KAM yovCav. th &Qa &7ch tf^g KA 
l6ov i6tl totg aTch t&v KM^ MA^ xal dLa th ^ri' tov 
7CQ(btov t&v 2JtoLxsCc3v ii 'bich KMA yc^vCa 6Qd"ij i6tLV' 
07CSQ idsL Sst^aL. 20 

19. MsCt^ov aQa xal yovCa 'fj i)7ch MKA xtL 
p. 162, 9] 5tL Ss ^ i^ch MKA trjg ^Tch SKN ^sC^mv 

i6tCVj SsC^O^SV tOVtOV thv tQ67COV' iTCsl ^Qd^oyovL^v 

i6tL tQCycjvov th KAM ^Qd^ijv i%ov tijv {)7ch KAM 
ycDvCav^ 6^std i6tLV {j {)7ch KMA' &6ts aiipista ij {)7ch 25 
KMS. d^pivycjvCov ohv tQLyovov tov KSM {} KS 

19. Rq (M^AFrsu, Vat. m. 2). 



9. vTtd] corr. ex &7t6 R. Tfjg] xov R. KA\ K e corr. B. 
12. AMA (alt.)] q, MAA RM. 14. r^?] q, roD B. 17. 
r«s] tov R. 18. KM'] KA R. 23. tovtov rbv rQ6mv] Br; 
ovtfog q. 24. tQiyfovov iati q. 26. K^M\ KM.^ <\s Jj 

Enclides, edd. Heiberg et HengQ. TIL. W 






•«-."H^-r" 



$ M 



Ifi^jL jr KX rf^ KJkL £X«£ «nrr rfcTon: aiscr d^^o- 
yMuc xa KS\* KMA ^^c? ijfxnt rc^ 2190«? mc^ 

2L .1/ '/^ytuaq^ ro £pe ^o riP^ KS ftfor fOTi Tof^ ^xo 

f, t(yv /T^. i?A'« ofu^tyz zex xo cao r^>? f^ i^9w roig 

(cxh r(rp Kyf. MA, mu itfti xa cxo xaw KS^ S\ 

fU,CiffV€c r(rv iaiti x&w K3i, MA- ij ycp SS rg MA 

l6Y^ l^\v i%2 XiigiuJLY^hr/gaumm ror MS aviSa iat- 

ivavtu^, Y^ dl KS rf^s KM ^i^av. jeci ro a^ iath 

rf^tf KS rffv ibw> rf^s KA lui^av' otfr* jeoi ^ KN 

rff^ KA {uCimf. Idiixfh; de xtd ij KS rfgS KM fisi- 

itov' Mtf dl Yf SS rf^ MA' iav aga rijv MA art rijy 

SN /jfaQfUi^toiiiv^ ivTog xeealrtu ro KMA rQiy&vov 

rfri) KSS rQtyufvov^ Tcal dia ro xa' rou a' rav Hroi- 

/» XfC^ fit^tiov iifruL fj vnb MKA rf^g vxb SKN' oxeQ 

f.dn dttl^ai, 

Ad prop. VLL 

20. rf.yQdtpd-co yuQ xegl xb tQiycyvov xvxXog^ Tcal 

ix(if.ftXri0i>(a0av ut KA^ KT ix" 
Mh.iug inl ric iV, S' ^ccl ixsl 

ic^ftkhtu dtCxvvtUL rj {mo Z^N 

i)g ixti)^ ovtJu^ fj (xQu &xh tov A 

tfi 7j/i XQog dQd^&g &yo^ivri 

t(ftUL ihg rj AA, xuXiv ixsl ufi- 
5 (ikhtu dfCxvvtuL fi r &g ixtbg 

yo. V fVai.(j, ]) in toxtu post prop. Vll); alia demonstratio 
UNi pro]». VII; cfr. oj)t. uet. 

ft. taov ftsrl Toftf q. 6. xa/ — 7. KM, MA] om. q. 7. 
MiN] IBIM i\. 10. fif/Jwr q. 13. ivtbg nsastrai] iyL- 

nifMTtn q. 18. ya^)) om. p. %v%Xog\ yiv%Xo V, corr. m. rec. 
8H. rt^»<^«^«) oomp. m, roc. V, ut p. 259, 1. 




\ 



IN OPnCORUM RECENSIONEM THEONIS. 259 

ov0a^ ij &Qa aich rov F XQbg dQd^o^g ayo^avrj l6taL G)g 
Tj FM, tovtov Sh ovtog i%6vtG}V SELxd"rj6£taL rj ZAN 
icsQLfpBQBva ^£c^(ov trjg tSjB TCEQiq^EQsCag ix tov naQa- 
XELiiEvov Ai^ftftaroff tov iv tp S' d^ECOQ^^^ati tov y 
j3fcj3Afcov tov UfpaLQLK&v ' t6ag yicQ 7C£QL(p£Q£Lag aq^aLQOv- 5 
6lv at xad^EtOL. &6t£ xal yovia 'fj 2J ^el^ov i6tl 
tfjg 0. o6t£ xal i] Z^ ^bl^ov (pav^^EtaL tr^g FB, 

21. Tb ax)tb d^EOQrnia Iv tL6L r«i/ &vtLyQ&(pov ^^ 
EVQrjtaL oiitog' t& t6a ^Eyid^rj i^cl trjg a^dtrjg Evd^ELag 
ovta xal fM^ i(p£^rig akXrjkoLg xEL^Eva avL6ov SLE^trj- 10 
xota rov o^^atog &VL6a (paivEtaL, 

l6to6av Svo ^Eyid^rj t& AB^ T/1 iicX trjg avtrjg 
Eid^ELag trjg A/1 ft^ i(p£%r[g aXMikoLg ovta xal avL6ov 
SLE6trix6ta anb tov Oftftarog rov E^ xal nQ067CL7Ct£to- 
6av axtlvEg al EA^ EA^ xal l6to ^el^ov fi EA tfjg 16 
E^^ xal dQd"^ ii i)7cb EAA. Uyo^ 5tL ij FA trjg AB 
^EL^ov (pavri6£taL, 7CQo67CL7Ctito6av axtlvEg aC EB^ EF^ 
xal ^CEQLyEyQd^pd^o 7C£qI tb AEA x^ixXog 6 AEA^ xal 
7CQ06£x^£^Xri6%^o6av a[ EB^ EF Eid^Etac i7cl tcc Z, if, 
xal avE6tdto6av a7cb rwv 5, F 6rj^£Lov tatg AB^ FA 20 
7CQbg dQd^ctg yovlag al BS^ FK, i7C£l oiv al AB^ FA 
l'6aL £l6Lv^ aklic xal aC BS^ FK^ cbg Sel^o^aev^ xal 
yovCa fj {)7cb AB& yovCcc tf^ i)7cb AFK i6tLV t6ri^ 

21. q, similiter M^RFu {tb r\' &XX(og M^); est opt. uet. 

prop. vn. \ 

4. Xri^xo^ V, corr. m. rec. Pro 8 — 11 M*Ru: %v xlcl 

T&v &vTiYQciq>G)v (fifra r^v 7tQ6ta6iv add. Ru) ix^i ^ rov d^say- 
Qritiarog ittd^seig iial dst^ig oijrtag (oQrcD Ru); iid. codd. ad 
jtOQQarsQco . . . rs&ivra add. yQ. nal (om. Ru) pbii iq>s^fjg kXU^ 
Xoig rs^ivra %a\ avicov Sisarriii6ra ro^ dpbpbarog avica tpaivsraL, 
12. ^ffrco 8vo taa MRFu. 19. ai — H] ratg EB, EF 

sifd^siaig sifd^stai ai BZ, TH MRFu. 22. slel q. 28. ij] 

rj MRFu. JFK] AFH Fu. iariv] om. MBPu. 



260 SCHOLIA 

xccl ^d^iq &Qa 'fj ccTcb rov A stcI tb ® rfi &nb tov A 
iitX xb K t6ri i6tCv' &6tE xal TtBQvtpiQBia fj AZ@ 
TCBQLfpBQBia tf^ KA i6tLV l'6rj, 'fj KA «pa %BQiq>iQBLa 
tfig AZ ^BL^c^v i6tCv, jroAAcf aga ^bC^cov tfjg AZ 
5 fj HKA. aXX" i%X ^\v r^g AZ piprjxBv ii 'bnb AEZ 
ycsvCa^ iTtl 81 trjg HKA iCBQLtpBQBCag piprjxBv ij imb 
HEA ycovCa' ycovCa aqa ii iitb HEA trjg iitb AEZ 
^bC^cjv i6tCv, (JAA' vnb ^lv trjg {)7tb AEZ ij AB 
B^d^Bta bgataL^ 'bitb 8\ trjg {)7tb HEA fj FA' [ibC^cdv 

10 ccQa &QataL ii TA tf{g AB, 

5tL Sh 'fj B® t6ri i6tl tfj FK^ SbC^o(abv ovt cag' 
iTtsl 'fj AB tfl FA t6rj i6tC^ xal xdd-Btoi ijtl tijv AA 
a[ 05, rK, TtaQaUrjkoC bC6lv al BS^ FK eb^BiaL' 
nQ06BKfikri%'Bv6aL TtaQakkrjXoL l6ovtaL, TtQO^BX^B^kif^^^^G}- 

15 6av xal i6tG)6av ai 00, KH^ xal BCkj^fpd^ca tb xivtQOv 
tov xvxkov xal i6tc3 tb P, xal catb tov P iitl ^bv 
tag 00, KH xd^Btov fjx%G}6av at PN^ P^, ijtl Sb 
tijv AA TtQbg dQd^dg rj P2J, 'fj P2J ccQa SC%a ti\v AA 
xatd tb U tB^Bt, dXXd xal 'fj AB tfj FA 'bjtdxBLtaL 

20 t6ri' xal koLTtij ccQa 'fj BS tfj UF t6rj i6tCv, dUd 
xal 'fj BU tfj NP t6rj i6tCv^ xal 'fj UT tfj PS t6rj 



1. icjto (pr,)] corr. ex V7t6 R. 3. tari ^^^^^ MRFu. rj] 
T§ Fu. 4. tfjg (pr.)] hinc fol. eodem uerso F, add. rov 0. 

iatl Fu. 7. HEJ (alt.)] HBJ Fu. 8. M (alt.)] om. 

MFu. 9. V7c6 (pr.)] ini Ru. V7c6 (alt.)] om. M. rj] 

rj svd-stcc MRFu. 10. oQ&tai] om. MRFu. AB]AB 

oQ&tai MRFu, 11. ^(frt] om. MRFu. 12. iati] om. MRFu. 

13. r^(pr.)] KF M, et corr. ex Fd u. slai q. r^(alt.)] 
KF MRFu. 14. TtQoasTipXrid^staccL — 18. Sixa] Sii^x^a} ndXiv 
dicc Tov TiivtQOv tov P TtQbg dQd^ccg tjj Ad i) P2, v.ccX dixoc &qoc 
MRFu. 19. Scjtonsitai u. 21. xat (pr.)] om. u. iativ] 

om. MRFu. tari iativ (alt.)] 7taQaXXriX6yQan^cc yccQ tcc BP, PF. 
%al ii NP &Qa rj NlS! tari MRFu. 



IN OPTICORUM RECENSIONEM THEONIS. 261 

ietCv. Kai sliSi TtQog d^d^&g xalg @0, KII' al 00, KU 
aQa t6ov aitixovCiv aito tov P, xal Std: tovto xac 
Bi6iv l'6aL, &6tB xal at 'fj^i^BcaL a^bt&v at SN^ KS 
t6av Bi6ivj S}v al BN^ rS l'6at' %al koiital ccQa ai 
05, Kr t6ai Bi6iv. 

Ad prop. Vm. 

22. 'Ev tp la d^BtoQT^^atL tov y' ^i^kiov t&v 2J(paL- 
QLX&v BVQifi^Bvg i^iod^Bv 6%6kiov^ 8 6v^^akBitai 6oi Big 
tijv 7CaQov6av 8Bii,tv. 

23. "i^iy 8b ii ^Z tfi BT' d)g aQa ij BT TtQog 10 
©Z, ovt(Dg i^ {)7tb ^KZ ycavia TCQog tijv iTtb BKF 
yoviav. &g Sh ij BF TtQbg tijv @Z, oiitog ri KF 
XQbg KZ Sioc tb tQiyovov tov KBF TtaQcc ^iav tov 
TtkBVQov ^xd^av tiiv &Z xal i^oyovia Blvai tic tQiyova. 

24 'T7tBQnB6Bltai tifv KZ p. 164, 12] d}g &7tb 16 
^Bi^ovog Sva6t7l^atog yQa(p6^Bvog^ OTtBQ i6tlv 'fj €>K' 
fiBi^(DV yaQ avtrj trjg KZ' &6tB 'b7tBQ%B6BttaL tijv KZ 
d)g ika66ova tijg K0. 

25. Ovt(Dg fi FK p. 164, 25] Svic tb i6oy6vLOv 
BLvaL tb BFK to 0ZK xal i%BLV &vdXoyov t&g ^tlBv- 20 
Qdg^ mg trjv BF TtQbg ti^v FK^ tijv ®Z TtQbg ti^v ZK. 

22. V^q (ad Sphaericoruin Theodosii m, 11 in iisdem codd. 

in mg. exteriore legitur lemma hoc : foro xqlyfavov dQd^oyAviov 

rb ABV, >tal TJx^o} xig ii A/i. d£r|at, Zxi i\ BF Ttgbg xiiv Bd 

iisltova X6yov i%Bi i]7tSQ ij 'bnb AJB ycavla Tegbg xi^v AFB). 

23. VVat.F(pquR). 24. q. 25. v^ 



\ 



1. al — 2. tcov] ag dsdsLyixccr teov &(^cc MRFu. 2. $u£] 
tcsqI MRFu. 3. tauL siclv MRFu. 4. BJV1 e corr. Bu. 

11. 0Z] xiiv eZ p. JKZ] e corr. q, KJZ Vp; Kd, ZJ .. 
RFu et eras. pr. J Vat. 12. yfoviccv] om. p. 13. K,1^^ 

xi}v XZ p. 14. slvoci] icxv p. ..^M 



262 SCHOLIA 

&6rB Tcal ivakk&i^ hg xiiv BF TtQog ti^ SZ^ xij^v FK 
TtQog rijv ZK. kXX' 6g fi BF TtQog ri^ ®Z, xal ^ 
^Z %Qog riiv SZ' t6ri y^Q ij ^dZ ry BF. &g aQa 
i\ /dZ TCQog rijv 0Z, ovr (Dg ij FK TCQog ri^v KZ. 

6 26. ^Slg yaQ at ycaviaL^ 8i mv oQ&vraL r& OQ&iuva^ 
i%ov6L TCQog alkrikag^ ovrcog xal ra OQihfuva dviL r&v 
ycovLmv XQog akkrjka i%£LV q^aCvovraL, cjg &Qa kotTtbv 
ij 2JP yovCa i'%SL TtQog ri^v P ycjvCav^ ovrog i%£LV 
(paCvaraL xal rb ^Z TtQog rb BF, i^ d^ ycovCa i^ 2]P 
10 XQog riiv P yavCav ikdrrova Xoyov i%SL fJTtSQ rb cat6- 
6rrjfia rb KF TtQog rb KZ. xal rb ^Z aQa TtQbg rb 
Br ^LXQdrsQov (paCvsraL TtaQoc rb KF TtQbg rb KZ. 

Ad prop. X. 

27. ''H%d^G) yo^Q Siic rov H 6rj(AsCov r^ BK TtaQ- 
15 dkkrjXog ii HE. insl ovv at bil^SLg nQorsQOv TtQbg rijv 

HE 7tQO0nC7trov6LV xara rd H^ A^ M ^rj^sta ijTtsQ 
TtQbg riiv KF^ xaC i6tL ^srscoQdrsQov rb H rov A^ rb 
8\ A rov M, xal dLa ^hv rov H 6rj^sCov ii BHF 
(psQsraL axrCg^ d^d 8i rov A r} BAZ^ SLa 81 rov M 
20 ii BM^j yLStscDQorsQa rj ^sv BF rfjg BZ^ ij 8s BZ 
trjg B^. 

28. Tb l' iv akXG) ovtGig* i^rcj yaQ H^fia rb B 
avc3 rov FK i7tL7ts8ov xsCfisvoVj d(p^ ov b^^arog 7tQ06- 
7tL7trsrc36av dxttvsg ai BF^ B^^ BZ^ BK^ av i] BK 

25 xdd^srog s6rc3 i^tl rb i^toxsC^svov i7tC7ts8ov. ksycj^ orL 
^rb Fjd rov AZ ^stsc^QotSQOv (paCvsraL^ rb 8% /iZ 

26. MVat.^Ru(F). 27. VVat.(q). 28. q. 



7. Xom^v^ X6yov Vat.^ 9. ii (pr.)] d Vat.\ 10. PJ 

O u. ^nsQ] sCnsQ Vat.\ 12. XZ] ^Z u. 15. t^v] •/• V. 



IN OPTICORUM RECENSIONEM THEONIS. 263 

tov ZK. eikifitpd^G} yaQ iTtl r^g ZK xv%ov 0Yj^stov 
ro E^ xttl ^O-co TCQog d^d^&g rfi ZK ii EH, xal btcsI 

aC OlpSLg TCQdtSQOV TtQOg tijV HE %Q067CC7CtOV6LV fpCSQ 

TCQog tijv Er^ 7CQo67CL7Ctst(o tfj HE fi ^sv Br xatoc 
tb H 6rj^stov^ ^1 Sh B^ xatcc to A^ ii S% BZ xat& 5 
tb M, S7csl ovv ro H tov A ^stsoQdtSQdv i6tt^ tb 
Ss A tov M, ^AA' iv o5 iett tb Jf, iv tovtfp tb P, 
iv ^ S\ ro A^ iv to^ito) ro ^, iv S Sl ro M, iv 
tovtfp tb Z, Sicc Ss t&v BF^ B^ fj ^F ^aCvstai^ Sii^ 
Ss tcbv B^, BZ ii Z^5 Slcc Sh t&v BZ, BK ij KZ^ 10 
ovxovv r/ ^lv Fjd r^g ^Z ^stsooQOtiQa (paCvstaL^ ij 
Sh ^Z trjg ZK' tcc ydcQ {)7cb ^sts(OQOtSQ(ov ixtCv(ov 
6Q(Ofisva fists(0Q6tSQa (paCvstaL. tG)v aQa xdt(o tov 
o^fiatog xsL^svcov xal tic s^rig» 

Ad prop. XI. 16 

29. HakLv iccv iydyrjg 7caQdkkrikov sid^stav Sl^: 
tov r, (pavsQbv i'6taL &7cb tcbv 6rifieC(ov, 

Ad prop. XII. 

30. Tovro d)g &7cb tov ff' g)avsQ(oteQOv yCvetai. 

Ad prop. XIV. 20 

31. ^AvtC6tQO(pov' ixet ^hv y&Q i)7cb tov onfiatog 
itedifj t& ^eyed^rj^ vvv Sh avc3 tov o^^atog. 

Ad prop. XVL 

32. ^AvtC6tQO(pov ^ d)g ei vorjd^eCrj tb ^x^fia ftera- 
tLd^e^evov avcD xdtcj, 86 

29. VVat.q. 30. VVat.q. 31. V». 32. V^ 



4. BT] B e corr. q. 21. itTtd] itTe&csqQv^ Y^ 



264 SCHOLIA 

Ad prop. XIX. 

33. 'EtcI ro B iciqas p. 176, 16] iistaxLvov^svov 
SrjkovdtL tJ tov svdxtQOv t) tov 6Q&vtog' ov yccQ Tcatic 
jtQihtrjv tv^bv impokijv tfjg o^fog xar' i(i(pa6LV dQcc- 

5 d"i]6stccL 7caQ& trjg oil^sog iv tp xatOTCtQp tb axQOv 
roi) vipovg, 

34. Ovtcog yccQ ivoQ&iisv t^ i667CtQCJ^ sc^g o^ tb 
axQOV iv ait^ tov dod^ivtog ^isyid^ovg tScDiisv. 

35. ^Ev totg KatOTCtQixoig p. 176, 18] (pri^l yicQ 
10 ixsL6s 6 EvxksLdrjg ovtcog' cc%b t&v iTCLTcidcov ivojttQcov 

xal xvQtcbv xal xoCXov al oipSLg iv l'6aLg yc3VLaLg ava- 

xk&vtaL, 

ccQiid^SL Sh aifta xal tb iv totg ^QOLg t&v Kat- 

OTCtQLXcbv siQrj^ivov iv67CtQOv tsd-ivtog iv iTCLJtidc) 

15 xal ta s^rjg. 

Ad prop. XXI. 

36. 'EvaQii6^cj yccQ iv rc3 iii6c) SLa^trnnaxL tcbv 
axtLVC3v ^iysd^og asl ivaQ^6^c3V^ sc^g oi Sloc tcov ccxqcjv 
avtov HSca ra axQa rov Sod^ivtog iisyid^ovg. 

20 Ad prop. XXII. 

37. Ox)Sh yocQ aiia ^XiicsL SAoi/, Zva 6vvaL6^ritaL 
C3g 7CSQLg)SQ0vg tov OQcofiivov. 



33. Vat. m. 2, rs. 34. VVat.RFp (qrstu). 35. V*. 

36. VVat.pr(q). 37. RF, Vat. m. 2, u(t). 



5. naQo] TtEQi r. t6 aiiQOv'} r, om. Vat.s. 6. vxl)Ovg'] 

(ijl)sa}g r. 7. a%6Xiov add. p. ovrcoff] ovrco ptR. i66nxQ(p\ 

7iat67tTQ(p p. 8. iv] corr. ex ^ m. 2 V. sfdoaiisv V. 18. 

de^] om! Vat.r. ivocQii6^a)v] om. r, lac. relicta. 19. sl'$(a V. 

21. 3Xov} (off F, om. Vat. 22. TtSQicpSQovg] TtsQKpsQslag Vat. 



IN OPTICORUM RECENSIONEM THEONIS. 



265 



38. 'Eccv iv x& ai)tm iTCLTCeda)^ iv c5 xal ro ofifia, 
xvTckov TceQifpiQBLa T£'&'^, ii xov yc&nXov TceQiipeQeia 
eid^eta yQaii(iii (paCvetai. 

^6x(o xihckov TceQLfpiQeia fj FB iv rp ai^xa iTCtTcidG) 
xeL^ivrj^ iv c5 xal xb o(i(ia xb A^ &(p^ oi ofifiaxog 5 
7CQ067CL7Cxix(o6av axxtveg al AB^ A/d^ AE^ AZ^ AH^ 
AS^ AF, Xiy(o^ oxl fj BF xiixkov 7ceQL(piQeLa eid-sta 
(paCvetaL, xeC^d^c^ trlg 7ceQL(peQeCag tb xivtQOv xal i6t(o 




xb Kj xal i^celeixd^^oCav eid^etaL aC KB^ K^^ KE^ 
KZ^ KH^ K@^ KF. i7cel ovv fi KB eid^eta imb t^g 10 
V7cb KAB ycovCag bQataL^ fj dh K^ {}7cb t%g {mb 
KA^., fj dh KE i^cb trjg V7cb KAE^ (leC^cov aQa (pa- 
v7J6etaL fj iihv KB trjg K^^ ij dh K^ trjg KE, 6fioC(og 
xal ix tov etiQov (liQOvg rj iihv KF fieC^cjv (pavrl^sta^ 
trjg KS^ fj d} KS tfjg KH. i7cel ohv tb aitb 6v(i' 18 
PaCveL^ S7ceQ &v xaC^ ei evd^sta {)7cixeLto i^ 7CSQLq)dQSia 
rj BF^ 6vvipaLve^ ro t6cg t6ag SrjlaSij dvCdovg (pat- 

38. MR(F, Vat. m. 2, Aqu). 



1. &XXa)g tov xy' i) ^sT^ig M, &XXa)g tb %P' q. idv\ Ubp 
ydg Vat. 1—3. om. Aq. 2. i^ — ^- «vx^o»] m. reo. IL 



8. xa^ey-a-ca] a/lifqp'^© q. 16. «H q, om. MR. ij BV 
g>SQSLa q. 7tSQi(psQSia] yfQvioc MFB. 






266 SCHOLTA 

v£6%^aL xal iieilovcc tiiv jtoQQfotiQO} eid^etav TCaQit tijv 
i^s^VS^ Bv^st^cc dfc<J: tovto i^ BZF q^avtd^stat neQi- 
(psQsia. 

dwatbv Sh tovto Sscxw^d^at xal hcl ttlg xoCXrig 
5 7t£Qiq}£QSLag. el yocQ ro K 'b%ot£%'BCifi th o(ifia ocal 
6rjii£tov tvxbv tb A ioctbg trig tov xihtXov Jt6Qiq>£Q£Cag^ 
£ita &'3tb tov A TCQbg tijv xvQti^v n£Qiq)iQ£Lav tov 
xvTckov ^id^^taL al AB^ AA^ AE^ AZ^ AH^ A®^ AF 
xal dxttv£g ccTtb tov K o(i(iatog htl t& B^ ^^ E^ Z^ 

10 if, 0, r 6rjii£ta^ t&v TtQog tijv xvQf^ ovv n£Qi- 
(piQ£Lav 7tQo67tL7ttov6&v £v%^£LGiv ila%C6t7i xal xatct 
(pavta6Cav hg xal xatdc aXif^^^^Lav ij iL£tai,i) xov te 
6riiL£Cov xal tfjg dLa(iitQOv 6Qad"iil6£taL^ t&v dh SXXcav 
a£l ij iyyLOV tfjg ika%C6trig tfjg &7tGit£QOV iXdttcav 

15 OQcctaLj Sij 6v(iPatvov bQcctaL^ xal £t7t£Q i] BZF 
7C£QL(piQ£La £v%^£ta i^tot^d^^Crj xal xa%^£tog iTt^ aiitijfv 
rj AZ' od^^v Sloc tovto xal (pavta6Cav ^vd^^Cag a7to^ 
0t£k£t ri 7t£QL(piQ£La^ xal [idkL^tc^ £t a7tb 7tk£Covog (paC- 
voLto SLa6tifiiLatog^ &6t£ (lii ^vvaL^d^dv^^d^aL iificcg tfjg 

20 xvQtdrrjtog. , 

Slcc tovto xal OL (lii Ttdvtc^g a7tot£ta(iivoL xdkoi 
ix 7tkayCov fiiv 6Q(hii£V0L a6xdka6iia ix£LV Soxov6lv^ 
iTtoxdtcod^^v Sh ^vd^^tg £LvaL^ xal at 6XLal Sh tcbv xqC- 
xcDv iv rc3 ait^ iTtLTciSco x^Lfiivc^v^ iv S xal tb o^fia^ 

25 ^id^^taL (paivovtaL, 



1. xat — noQQOiTBQGi^ om. lac. rel. Vat. fqv (alt.) — 2. 
Sia,] q, tfig icp' rig rb (dem. ras. M, spat. 2 litt. E) iari MFRu. 

6. rfjg 7C£QV(pSQsiag rov HVTiXov MR. 9. rcc\ supra scr. R. 

E] corr. ex K R. 10. r&v] hinc etiam r. ovv] q, om. 
MR. 11. Tial] om. r. 15. 8 — dQ&raC] postea ins. q. 16. 
n8Qt(piQSia] ycovicc R, om. M, yoDvia tov hv-aXov r. 21 — 25. 
om. A. 22. Scr. iyxdXaa^ia. 



IN OPTICORUM RECENSIONEM THEONIS. 267 

Ad prop. XXin. 

39. noL7J66L drl p. 180, 22] 8l& to tcqcjtov tmv 

UcpaLQLXCJV. 

40. 'EipAiljovxaL at BA^ BZ ^, 182, 2] ^'^^^ra^ 
ov6aL al aocttvag tcbv 6Q(hvt(DV tijv ^cpatQav, 5 

41. Kal iTCsl ixdetrj xtX, p. 182, 5] ixd^trjv t&v 
TCQog tp & ymvL&v dQd^fjv 6vvd^ov6Lv elvaL aXXoL iilv 
a^cog akXcog^ iyh 8\ tovtov tbv tQ^TCOv, iTCsl sxa^tov 
tcbv KZB^ KAB 'fjiiLxtixhdv i^tLV^ ij KZB nsQLq)iQBLa 
l'6rj i6tl tfi KAB TCBQLtpSQsCa^ i)v ij KZ t6ri tfi KA' 10 
t6aL yccQ svd^staL al KZ^ KA ix tov xivtQOv oi)6aL 
rov Zr^ xvxXov {)7C0tSLV0v6LV aitdg' XoLTcii &Qa ii 
ZB JcsQLg)iQSLa r?J AB 7CSQLq>BQSLCc t6rj i6tCv, &6tB 
xal sid^sta rj ZB tfj BA t6ri i6tCv. ijcsl oiv Siio 
tQCycovd i6tL td KZB^ KAB tdg dvo jcXsvQdg tctg 16 
ZK^ KB tatg Svo nXsvQatg tatg AK^ KB t6ag ix^vta 
xal tiiv pd6LV tijv ZB tfj pd6BL tfj AB t6rjv^ xal ri)v 
ycovCav tijv iTcb ZKB tfj ycovCa trj ijcb AKB t6riv 
B%BL. TcdXLV iTCsl d^io tgCyova td ZKS^ AK& t&g 
Siio TcXsvQag tag ZK^ KS tatg Sio TcXsvQatg tatg 20 
AK^ KS t6ag B%ovta xal f^v ycovCav tfi yovCa^ xal 
tijv pd6LV tijv Z® pd6SL tfj 0A t6rjv b^bl, xal ixsl 
Bvd^std tLg SlA tov xivtQOv fj KB sid^stdv tLva fii^ 
SlA rov xivtQOv tijv ZA SC%a tiiivBL^ xal JCQbg dQd^icg 
avtijv tsiiBt, 85 

« 

39. VVat.u. 40. VVat.RF. 41. ME(Vat. m. 2, Frs). 



5. al ScTiTtvsg] om. R. 8. rdv] RF, om.MVat. 9. KAS] 
JTZ^ MR. 10. tari — ^sgttpsQsLoi] R, om. V. JL7. %al(jpr,)\ 
^sv M. 23. TiSvtQov] X RF. 24. yisvtQov] % a. ZA} 

AZ s. 25. TS^vy r. .^ 



268 SCHOLIA 

42. ^icc xh TtccQccXXrjXov p. 182, 6] TCccQdXXriXog Sict 
tov xrj' xov a twv 2koix£C(ov, 

43. "Atcbq ii, avdyxrjg (pv6txf}g iicl t&v 6Qa}[ievov 
yCvBtai^ tavtu xal Sl' a7CoSsC^sc3v TCL^tdtCaCd^ac /Jov- 

5 Xdiisvog 6 yscaiistQrjg t&v d^scjQrjiidtcov JcaQafivd^Cag dscb 
tG)v yQa(i(i&v xofiC^SL x^dxXovg dvayQdfpav iv tatg dich 
tcbv dfiiidtov d7CO7CS(i7C0iisvaLg dxtt^Lv xal sicCicsSa SlA 
tG)v oil^sov ixpdkkc3v xal stsQa tOLavta tcolov^ 0^% 
StLj tavta iihv idv yivrjtaL^ s6taL dXrjd^^g i^ tOLdSs 

10 a^dtov TCQota^Lg^ xal xad'^ bv avt6g g^rj^L tQOTCov d^so)- 
Qrl6ov6L To toL6vSs (fxw^ ^^ oipsLg^ idv Sh (lii yivrjtai^ 
il^svSrjg' fj ydQ av^ si tovto ovtcog sl%sv^ iv tfi XLd^cc- 
v6tritL tCbv djcoSsC^sov ixsLto av 'fj tovtov svQS^vg 
fi6vov^ aAA' oix iv tfj (piicsL rwi/ bQo^Livov^ xal yQa- 

15 (foiLSvov il\v rwv xvxkov t) rwi/ iiCLTciSov ixfiaXko- 
liivov ioQccto av tb bQo^svov^ og b EdxksCSrjg g)rj6Cv^ 
li'^ yLVOfiivov Ss rovrov ovx av id^soQstto tOLOvtov^ 
d}g slvaL iiakkov aith Slcc tijv d7c6SsL^LV oiitog s^ov 
fj Slu tijv {pv6LV, th S^ oi)% ovtog ix^L^ dkld otcsq 

20 i^ dvdyxrjg q^v^Lxrjg avit^aCvsL Tcd^x^^'^ i^^^S oiIjs0l 7Cqo6- 
Pakkov6aLg to tOLoSs ^x^JJlicctL olov xvkLvSQOSLSst -^ 
xovosLSst t) 6(paLQ0SLSst inl %Xiov dq)L6taiiivaLg tJ 
^Q06syyL^ov6aLg aito^ tovto Sij fiovk^iisvog dico- 
SsLXVvsLv b ysofiitQrjg jcaQafivd^sttaL tijv d7c6SsL^LV S^d 



42. FVat. 43. V* ad prop. 24, p in textu inter propp. 23 
et 24. 



1. naQdXXriXos] om. F. 2. xrj' — DvoixsicDv] xa' r&v 

E^tlXsISov F. 4. yivGivxai p. ^ovXoiLsvog] povXstat p. 5. 

t&v d'sa)Qriiidta)v'\ supra scr. V. 6. xo^/fcov p. 7. &'iitZ6i, p. 

13. tovtoDV^ t&v toLOvtcDv p. 20. 7CQoa§aXovGaLg p. 



IN OPTICORUM RECENSIONEM THEONIS. 269 

imTteScov xb xal xvxXcov xal tOLo^drov tlv&v^ Tva ocatoc 
ndvta ^iiiKpcsvov avxiiv 7C0L7]6rj rotg iv rfj yE(x)iistQLa 
CtOLXSLOLg xal 7CaQa6xsvd6rj tbv axQoatijv fistcc TColkrjg 
otL iLdkL6ta ijSovf}g iyx^&JCtSLV toXg d^soQtlfia^LV^ &6%sq 
d^sksL xal ijtl tfjg aQLd^iirjtLXTJg l6rLV ISslv aitbv tcol- 6 
ovvta xal ysmiistQLag xal t&v aXkov (lad^rjiidtcjv. otL 
(ihv yccQ Svo rsrgayc^vcov aQLd^iiov slg iis6ov avdXoy6v 
i6rLv dQLd^iidg^ rovro dXrjd^sg i6rLV^ dXT^ oi) Ssl rovro 
li6vov diC airrjg slSsvaL rfjg al^d^ifj^sog^ tv' ovrcog 
sHtcco^ dXkd xal Sl' dxoSsL^scag d6(paXs6riQav s%slv rijv 10 
tcsqX avrov yvcb6LV, bfiOLcag S^ xal rovro dkrj^ig i6rLV^ 
8rt, idv Svo svd^staL riiivc36Lv dkkrjkag^ rdg xard xoqv- 
q)ijv ycovCag l'6ag dkkifjkaLg %OLTfj6ov6LV^ xal ^avsQbv 
dnb rfjg al6%^rj6sog^ dkX' oix d7c6%Qrj %Qbg i7CL6rifjiLrjv \. 
rb ovrog slSivaL [i^vov^ dkk^ Exslv rovro o^okoyoii- 16 
lisvov Ix rLvov TCQoriQcav xal yvoQLitcoriQcav rovro 
Si i6rLv rj d7c6SsL^Lg. 6 avrbg roCvvv k6yog i6rl xal 
iicl rovrov^ Zrc q)v6Lxog Sxsl fj SQa6Lg ovrog Sqccv rd 
oQOfisva^ &g 6 EixksCSrjg (prj6Cv^ Tva Ss xal i%L6rifjiLrjv 
a-^r^v ^';|ra9fi-£i/, %Qbg xardkrjil^LV dxQLfis6riQav TCaQa- 20 
kaii^dvovraL iv ratg dnoSsC^s^Lv aircbv x^dxkot xal 
ijcCjcsSa xal akka roLavra. 

XQfj Sh slSivaL^ og rovg xiixkovg xal rd iTcCicsSa^ 
orav ^sv 6qo^sv avrd rd 66fiara olov 6q)atQav r^ 
xiikLvSQOVj vorjrog Sst dvayQdtpsLv t) ixfidkksLV^ 5rav 26 
Sh iv iTCLTciSo^ al6%^rjrog d)g ivravd^a. 



4. iyyvitxuv V. 6. ^iad^ri^tLTi&v p. 7. fiieov] eic 

Vp. 10. $l'] Sid p. 13. TeoLi^aovGi p. 16. &XV] kVA p. 
24. oiov — 25. %vXiv8qov] ins. ead. man. V. 



270 SCHOOA 

Ad prop. XHV. 

44. Ai PZ^ PZ xa»' h^ iffaxzwtai p. 184, 5] 
itpdxxovxiu oQa dia xo iv xa is' xov y' xav JExoi%si&v 

XOQUflUC 

5 Ad prop. XXVL 

45. ^OfiiiaTGfv dia6xr^na xb BT p. 186, 7] x^ d\ 
voatv, 5x1, fi dui6xa6i,g xGrv 6fifufrc9tr xaQaUir^log i6xi 
rg ducjUxQa xov xvxkov» 

46. Tovxi6xtv hcL^x^eufdnf axb xov K ixl xa 
10 B^ r 6riiuta av^SLGjv. 

47. "Ela^^ov av etri p. 186, 16] ai yoQ xBd^sirj xb 
iftiuc ixl tov 0y duc xb xy' xav "Oxxixav ilatxov i^iii' 
^qxagiov iHp^rfisxcu vxb xov ivbg ofifcoro^. 

48. Tb ZNjd p. 186, 16] xovxi6xLv xb vxb xov 
15 xvxXov diOQii6(uvov xov xsqI xijfv ^NZ. 

Ad prop. XXVIIL 

49. 'X)v TQOJtov inl xov xy' xal x8' iSai^sv^ ovxag 
xal inl X(bv 8vo rovxav xov xr{ xal xO"', xk-^ ixst 
liav iicl 6(paCQag^ c)8a 81 ijtl xvXlv8qov. 

20 Ad prop. XXX. 

50. KvxXov a%ovxog xij[v fid^cv p. 192, 2] ovxi^ 
8v6xi a6XL xig x&vog iiri Sxcov xvxXov xijv fid^cv^ tovx6 
q)ri6LV^ &Xka xr^v (pv6vv xov xd)vov %aQa6xffiai ^ov- 
Xdiiavog. 

44. VatRFu. 45. Vq. 46. VVatq; quid sibi uelit, 
nescio. 47. Vat.RF. 48. VVat.q. 49. V»q. 50. R 

(VatAFq). 

3. iv t£] supra scr. R. 14. ro] supra scr. m. rec. V. 15. 
~ comp. Y, TCtxQd q. JNZ] ANZ VVat.q. 17. ovtci} q. 



IN OPTICORUM RECENSIONEM THEONIS. 271 

51. Kal iTcl xoijtov xal tov iiata tovto 6(iOLa ri 
dst^ig Tckijv iTcl xd>vov, 

Ad prop. XXXn. 

52. Ta 7tQOV7tox£L^evc) imTteSm p. 194, 19] rovr- 
e6tc tfi pd^ec tov xiovov. 5 

53. Ovxovv 0v(i7C£0ettai p. 196, 2] iiceiSii xarc!: to 
ai)tb axQOv av(o (ilv xard: ro 5, xdto) Sh xatct tb K 
6vvd7Ctovtac. 

Ad prop. XXXTTT. 

54. "EXa66ov q^aivetai p. 196, 22] yg. ^et^ov ^lv 10 
i^tac tov xG)vov t6 OQdifievoVj ^Aarrov Sh q^aCvetai^ 
taneivoteQOv S^k iXa66ov (ihv i6tai^ S6i,ei S\ (let^ov 
(paCve^^^ai, 

55. Tovte6tiv Xva iicC tcvog eid^eCag tb ofiim ij, 
^tig TCaQdkkrjXdg i6tc tfj dicb tfig xoQvg)fjg tov xd)vov 16 
TCQbg tijv 7ceQi^)iQeiav avrov dyoiievrj eid^eCa. 

56. '760V Si tb (liv JCQbg rc5 N xtL p. 198, 9] 
iav ydQ^ xad^hg efQrjtac iv ta Xa d-ecjQiliiatL^ dicb 
rov N oii^atog 7CQO07ce6(o6LV dxttveg 7CQbg tijv tov 
XG)vov 7ceQL(p£QeLav d)g aC NT^ NO^ xal aTcb t&v T, 20 
i7cl tijv xoQVfpijv f^v ^ i7CLt,ev%%^G)6Lv eifd^etaL Sg aC 
T^, 0^, tb SlA t&v NT^ TJ iyeCTceSov xal tb Scd 
tcbv NOj O^ xoLvijv rofti)v e%eL tijv ^iV, iq)* 'fjg idv 
ted^fj tb oii(ia d)g xarcJ: ro N xal ro 0, t6ov del tov 



51. Y\ 52. VRVat.u. 53. VRVat.M^qtu. 54. 

Vat.RMist. 55. VR(Vat.qrstu). 56. VR(Vat.MAqrstu). 



12. rccytsLvonrsQov Vat.s. Ss (pr.)] om. Vat. ^^'«£^1 

om. V. 15. TeaQdXXriXdg] = R, 6q&i/j s. 16. sif&si^'] om. VjH 

18. ydQ] om. Mt. Xa'] V, Xy' m. rec. ^O. x&wX-w*^^ 



272 SCHOLIA 

TCfovav rb oQWiUvav inp^fiEtia dia xo Xa ^e&^iuc' 
6iJLOi€9g 7ud ixl XY^g AH, 

Ad prop. XXXV. 

57. 'Op^ av tlr^ p. 200, 23] ix£i yhq l6ri i6xlv 
5 Vi KJB yayvta xfj KB^, ij di KZB rg ZBK^ dvo 

aQu at B^K, ^ZB 8vo xalg ZBK, KB^ t6€a si6Cv, 
S)6xe al xa66aQ£g at BZK^ Z/^B^ /IBK^ KBZ dvo 
x&v jdBK^ KBZ^ XOVXS6XL xf^g jdBZ^ dvxlMSCovig 
al6Lv. iXXa al xi66aQ£g 8iio dQd^atg t6cu si6Cv' iv xm 
10 XQiytavfp yaQ el6i xa ^ZB. &6xb fi jdBZ yovCa 
dQd-ij i6xvv, 

58. ^Exsl yicQ at XQstg t6(u ai^lv at jdK, KZ^ KB, 
6 ccQa xivxQp xa K^ 8ia6xriyLaxL Se xm Kjd xvxXog 
yQucpdiLBvog r^i^si xal 8ca x(bv B^ Z. &6t£ 6q^ i^ vxb 

15 zJBZ' iv tiiicxvxXCg} yccQ. 

59. 8 yccQ eC6cv o)g xf^g vtco ^BZ ^LaLQOviuvr^g* 
STtal cog iv XQiydyvcp XQalg al6cv. Sca xovxo xal 8vo 
oQd^atg i6aL' &6xa fj vTcb /JBZ 6Qd7J i6xL^ 8l6xl 8 
icpdvrfiav iv r« XQcycbvcp^ xal avxrj ag 8lg kafi^avo- 

20 icavrj 6Qd"i^ icxLv. 

60. At ScdiLaxQOL t6aL p. 202, 5] 8fikov 8ij oxl 
ov 7ta6aL 7cd6aLg al 8LdfiaxQ0L t6aL cpavifi^ovxaL^ dXXa 

57. VR(Vat.MFAqrstu). 58. VR(Vat.MFqrtu); 

eodem pertinet, quo nr. 57. 59. V* (ad riaauQSs lin. 7). 

60. V(Vat.pqr). 

1. ^ta — d-smQTifJLa] om. A. la'] mut. in ZjS' m. rec. V. 

2. diioiag — A2] om. s. 5. KBJ] KBF V, KJB R. 6. 

JZB] J supra scr. V. KBJ — 11. iativ] om. A. 12. 

ydg] om. t. Post tam ras. 2 litt. V. 14. tdtv] rov R. 

16. JBZ] J Z dirempt. spat. 1 litt. R. 22. naGULg al] 

V. 



IN OPTICORUM RECENSIONEM THEONIS. 



273 



liicc [iLa^ olov tfj EF fj ^B* ccvtrj yuQ (idvrj dvvatac 
i'(fag ycovCag tcsqlsxblv iLStcc ti]g AZ tatg TCaQLexofievaLg 
iicb tfjg A7j xal EF' tovto $} Sloc tb ^^ slvaL JtQbg 
dQd^ocg t(p {)7C0X8L(idvp i%L7ciS(p tiiv ZA, 

Ad prop. XXXVI. 5 

61. /:lLri%%^(o yccQ p. 204, 17] fti) TCQbg d^d^ag o^<ya 
SrjkovdtL r?J FA. 

62. 'H rz p. 204, 19] ov0a SrjXovdtL tov xvxkov, 

63. AfiiL[ia, 

TC&g Sh XQ'^ ^gbg dQd^ccg &yayElv tfj xsxlLiievri 10 
eid^eCa TCQbg tb iiCLTceSov (liav eifd^etav iv t& tov xv- 
xlov iTCLTciSG); oi) ydcQ xal eti^av Svvatdv iTCOxeL^d^G} 

y&Q tb 0XW^^ ^^^ ^^^ 
tov B ijcl tb iiCLjceSov 

xdd^etog i^x^^ 'h ^^j ^^ ^^ 
iTce^evx^cD fj AA. q>a- 
veQ6v^ StL 'fj AA iv 
t& tov xiixkov i%LniSip 

i6tLV. ^x^^ ^'^ ^^^ 
Tov A tfl AA TCQbg dQd-ccg fj AM' ^^ec S'^ iv tp avtp 20 
iTCLTciSG)^ iv ^ xal i^ AA^ tovti6tLv iv t& xtixXtp. 
iicel oiv ii BA dQd"il i6tL TCQbg tb tov xiixXov inC- 
TceSov^ xal nccvta &Qa tdc Sl& trjg BA ixCxeSa dQd^d 




61. RFMi. 62. Rt. 63. VR(Vat.MFqrstu); ad 

p. 204, 1: TJx&a) yccQ 17 pbkv TZ yxX. 



2. T^s] p, et corr. m. rec. ex ttJi/ V. 9. ^^ftfMK] Vq, 

om. cett. 10. Hl om. Mt. tisKXriiievT] V, sed corr. 20. 

Sij] e corr. V. 22. BA] B e corr. V. ' 6^9"^ teti ^. 23. 

6^^« iaxi] compp. V, J^ea stai R. * ^ 
Euclides, edd. Heiberg etMeuge. TXI. " "SA 



274 SCHOUA 

i6ti XQog xov xwtkov. h/ dl t&v dut tf^g BA ixi- 
niS(ov i6tl tb BAA tQiy covov' xal tb BAA &Qa tqC- 
ycyvov dgd^dv i6tL xgbg tb tov xihiXov iTcCneSw. xal 
tfi xotrnj tcbv imxidcjv TtQbg d^d^icg ^tai fi AM iv 
6 t^ tov xvxXov imjcidc)' ri AM &Qa nQbg tb BAA 
iTCLTCsdov dQd^TJ i6tiv. xal TCQbg ndeag &Qa tag cacto- 
liivag ainfig^ ov6ag S\ iv ta BAA ijctTciSc)^ dQdn] i6tiv 
7] MA. &6t6 xal TCQbg tijv AB dQd^ i6tiv. 

64. Kal avtri iihv fj aTcdSeL^tg^ el inf^te TCQbg dQd^&g 

10 rj EA tfl FA SLax^fj' t&ce y&Q &7cb tov F inl tijv 
AE evd^etav Swdiied^a xdd^etov ayayelv ti(v FZ^ Tcal 
ovtcjg fi aTcdSeL^Lg 7CQ0%(OQeT. ei Sh fi EA xdd^etog 
ijcl tijv FA Stax^^ Sei%%^if^6etai ndXiv fi vnb BAF 
y(ovCa trjg fj%b BAE iXdttcov tovtov tbv tQdjcov iicel 

16 fi BF dQd^TJ i6ti nQbg ro ijcoxeC^evov ijcCjceSov^ xal 
Tcdvta &Qa td^ Sl^ avtfjg iTcC^ceSa t^ ait^ imTciSc) 
TCQog dQd^dg S6tai. &6te xal tb BFA tQCymvov tm 
EA xvxkc) TCQbg dQd^dg e6taL. inel ovv ro FAB tQC- 
yovov tfp xvxkci TCQog OQd^dg i6tL xal tfj xoLvfj avtcbv 

20 tofifj fj EA iv ivl tcbv ijCLTciSov^ fj EA &Qa xal ta 
ABF tQLyihvG) TCQog OQd^dg i6taL' xal TCQbg 7cd6ag &Qa 
tdg dntoiiivag avtfjg evd^eCag xal ov6ag iv t& ijco- 
xeL^ivc) ijCLTciSc) t(p ABF bQd^dg 7C0L7]6eL yovCag. aic- 
tetaL Se avtfjg xal fj BA* xal TCQbg &Qa t^v BA 

25 oQd^ijv TCOLt^^eL yovCav. dQd^i^ &Qa f] f)7cb BAE' d^eta 

64. R(Mtu); ad p. 204, 11: ^Gto) TivTiXog, ov ^evTQOv rb A %xX. 



1. x&v'\ corr. ex ro5 m. rec. V. iitmBdto V, corr. m. 

rec. 3. dQ^Qv] icov R. 4. d^O-ag] t6ug R. ' 7. 8s] om. R. 

6^%"^] i'67i R. 8. i6tL R. 12. si] 7] Ru. 17 (alt.)J om. u. 

13. 8i8ax&ji u. 19. to5] ro5 E/i u. 23. rw ABF] supra 

scr. R. 



IN OPTICORUM RECENSIONEM THEONIS. 216^ 

Sh fj V7tb BAF. ikdxxGiv ccga i^ vTcb BAF xrjq v%b 
BAE. 

65. 'AvccTtakiv aqa p. 206, 26] BUsiSii bIubv &vd- 
TcaXiv aQa 7] ZA TtQbg xijv AB ikd66ova X6yov i%ei^ 
oh S%Bi fj FA TCQbg AB^ i^xiov xovxo^ oxl ijtl (ihv 5 
xrjg xavxdxrjxog x&v k6y(ov ndvxa 0G3t,Bxai Kal xb iv- 
akkdci, xal xb ^vvd^ivxL xal xb Suk^vxc xal xb dva- 
6XQiilfavxi xal xb dvdTcakiv^ olov dtg x6Sb TCQbg x6Sb^ 
ovxcog xoSb TCQbg x6Sb' ivakXd^ Sg x6Sb TCQbg x6Sb^ 
ovxog xoSb TCQbg x6Sb' TcdXiv d)g x6Ss TCQbg x6Sb^ ovxcog 10 
x6Sb TCQbg x6Sb' ^vvd^ivxc d)g x6Sb TCQbg x6Sbj ovxmg 
x6Sb TCQbg x6Sb' biiOLcog xal iicl x&v dXkcjv, inl S\ 
xfjg ixBQ6xrjxog xmv k6yc3v ndvxa [ibv xd &XXa 6Git,Bxai,^ 
xb Sb dva^XQi-^avxL xal xb dvdjcaXLV oixixL^ olov iiCBX 
x6Sb TCQbg x6Ss (isi^ova X6yov i%BL Vitcbq x6Sb TCQbg tjt^ 
x6Sb^ ivaXXd^ x6Sb aQa TCQbg x6Sb (iSL^ova X6yov S%bi, 
i^jtBQ x6Sb ^Qbg x6Sb' biLoCog xal i%l xov ^wd^ivxv 
xal SlbX6vxl, inl Sh xov dvxL6xQi^avxL xal xov dvd- 
TtaXiv oixixL^ dXXd xb ivavxiov yCvBxaL ovxcog' x6Sb 
TCQbg x6Sb (isC^ova X6yov i%BL VjictQ x6Sb %Qbg x6Ss' dvd- 20 
TCaXLV x6Sb ccQa TCQbg x6Sb iXdxxova X6yov i%BL i^TCSQ rdtfiL 
TCQbg x6Sbj d)g wSs sItcbv* ravra Sl 6 ^Hqcov SLaQd-QoLy.! 

66. Tb ydQ avxb fj ZA TCQbg xb BXa66ov iisC^ovcc 
X6yov B%BL ^TtSQ TtQbg xb iist^ov xb AB. 

65. VE(Vat.MAqu). 66. VE(PquVat.»). 



6. Ttdvtoiv R. 7. t6 (pr.)] mut. in rc5 R, ro5 V. r6 (sec.)] 
ta R, et V, sed corr. to (tert.)] trc5 V et corr. ex t6 R. 9. 
ivocXXd^ — 11. t68B (sec.)V om. R.' 13. tf^g'] tfjg t&v V. 

14. t6 (pr.)] corr. ex t& V. ovxcrt] -iti in ras. V. 22. 
oig — dittQd^Qol^ om. A. mg'] V, om. RMu et lac. rel. Vat. 

''HQGiv'] VVat., om. Mu et lac. rel. R. 23. Supra scr. dik 
tb {tov m. rec.) f tov s' E^xAa^dov V. 



276 SCHOLIA 

67. IlQbg dh tijv AB tvxov6av p. 208, 14] xal 
TtQog aitriv yocQ dQd^ag jcouiv oi Svvatai^ insvSi/i^ i&v 
svd^sta dtio sid^siaig ts^vov^atg dXXnlkag TCQog dQd^&g 
iTcl tfig Tcoivfig toiir}g i7tt6tadi}j xal t& di^ aift&v ini- 

6 TciSip TCQbg dQd^dg i6tiv' i)7c6xsctaL dh ain^ /Lti) ot)6a 
TCQbg dQd^dg iv tp Ag'. 

68. Kal Tcdvta &Qa xtL p. 208, 17] dtd tb d' xal 
tb vYi t&v UtSQS&v tov a ^i^Xlov. 

69. 'EscX tijv xoivijv aQa p. 208, 21] i%oiLSV yitQ 
10 iv totg UtSQSotg ^'SfhQriiLa' idv iTCLTCsdov XQbg ijCLTCsSov 

dQd^bv ^5 xal d7c6 tvvog erjfisiov avtmv iv svl t&v 
imTcidcjv iTcl tb stSQOv ijciTCsdov xdd^stog dx^^ i^l 
trjg xoivfig toiirjg %s6sttai tS)v imTcidoDV, 

70. 'H N3 iiSLtG)v p. 210, 4] dL^tv t^rj i)%sti^ri 
15 r?J EZ TJJ x)7C0tsd's^6rj fiSL^ovL t&v ix tov xivtQOv^ 

xal idv i} EZ iis^^ov^ xal avtrj d)g t6ri tavtrj. 

71. 'H NO p. 210, 11] -^ iVO yccQ ixtbg 7Cs6sttaL 
Toi) AUM tfiTliiatog' rj yaQ N3 tfjg NP iisC^Gyv i6tLV' 
iicl yccQ tfjg NS i6tL t6 xivtQOv tov xvxkov tov AU' 

20 fisL^ov ydQ i6tL tfjg AN. insl yicQ iv xvxXo) to 
ASM svd^std tLg i^ NS sv^stdv tLva tijv AM SL%a 
xal TCQbg dQd^ag ti^vsL^ iicl trjg NS aQa i6tl rb xiv- 
tQov tov ASM xvxlov, {)7c6xsLtaL Sh i} NS ^si^cov 
tfjg ix tov xivtQOv^ iicsLSii xal ij EZ^ xal dsl i} iyytov 

25 tov xivtQOv tfjg dicotSQOv ^sl^cov. 

67. VR(MFVat.Aqru). 68. VRVat.q. 69. VR 

(Vat.MAFqu). 70. V^ 71. VR(MFVat.qru). 

« 

6. iv] cbff iv A. 8. wj'] rj' R. 10. StSQSotg^ om. lac. 
rel. Vat. d-sagi^fjLaaLv Fu. 11. avr&v — r&v] in ras. V. 
13. r&v iTtiTtsSav TtsaslrocL A. 19. iTtH iitsi r, et V, sed 
corr. iV^] ^ in ras. V. 24. EZ] Z in ras. V. 



IN OPTICORUM RECENSIONEM THEONIS. 277 

72. ''Ert 7tEC6%^(D tfi i)%o tcbv EZK p. 210, 14] ^ 
yccQ vTtb t&v EZK iSetx^V i^ccT^tmv Jtaecbv t&v Sict 
tov Z SLayofieviDv xal %olov0g)v TCQog tfj jiB ycovvag. 

73. MeCt,(ov d^ i^ ^ P- 212, 1] tQLyfhvov yicQ tov 
APII ixtdg i6tL^ xal 'fj JtQog t& O ccQa ^eC^cov i6tl 5 
tfjg TCQog ta U, xaC i6tL fi [ihv sCQog tw O i'6rj tfj 
V7tb HEe^ ii Sl jtQbg t& 11 l6ri tfj vnb AEB. 

Ad prop. XXXVIIL 

74. Tov Hiiiiatog inl roi> xsvtQov tov xvxXov 
x£L(ievov. 10 

75. ^O^oCcog Se^ xav anb tov F xevtQov TtQbg dQd^ag 
ava^tad^rj evd^eta^ iitl Sh ta^itrjg tb 6(ifia tedij^ xal 
listaxLvfjtaL TO bQih^evov (leyed^og xatcc tfjg tov xtixkov 
jtSQLcpeQeCag TtaQakkrjXov ov t^ evd^eCa^ icp^ 'Ijg tb lififia^ 
i!6ov ael tb OQchiievov dcpd^i/j^etaL. 15 

Ad prop. XL. 

76. AeycD^ otL rj AB xtL p. 220, 2] tovti6tLV' 
Stccv i] AZ tijv d^i^LV iv r« xvx^p tavtrjv 6%oCri^ 
iXdttcjv 6(p%"ifi6etaL ^iteQ^ or^ fjv ava6ta6a jiij TtQbg 
dQd^dg. 20 



72. Wat. 73. x m. 2, m. 1 in textu inter iTtt- et 
-isvyvviisvri p. 212, 5 (del. m. 2 et in mg. coU.). 74. RVat.; 
cfr. p. 216, 4 not. crit. 75. Rur(M^Ft). 76. V(RVat.M^ 
APqut). 



3. dLayoiLSvav^ corr. ex diayiovlav V. 5. tj TtQdg] 0, 

iTCsl X. 6. iatt] ds? X. 7. HE@] r&v E@H x. 9. 

Tov v.svTQOv Tov xvx^ov] Tov liSVTQov R. 11. T] R, om. ru. 

12. Tsbiij] ^staTsd^fj u. 13. iisTcc%ivslTOii Ru. Tov\ om. u, 

19. oi^pQ^riGSTaL V, sed corr. 



278 SCHOLIA 

77. ^^Aoi/, Sr^ 7CQ6t6Qov Sst Sst^av p. 220, 12] sl 
yccQ xovxo SBixQ-fi^ Srt iXd66(ov ij imh BEA yiovCa 
r^ff ijco Tj^A ycoviagj yv(OQc^ov tb ^rjtov^evov Sg 

SiCt t&V 0QC3V. 

s 78. 'AXkct Sii l6tGi p. 222, 21] i%El eIubv^ Ztr i^toL 
S% fi ^Z ^LBit^cov tfjg Bx tov xBvtQOv r) t6ri i) sXd66G)v^ 

^vTtid^Bto Sh aitiiv (iBC^ova xal ISbl^b to AB (liyBd^og 
tov AZi ika66ov^ vvv iTtotid^BtaL tfjv ^Z t6r]v tfi ix 
tov xivtQOv xal SbCxvv6l ndkiv th AB ^iiyBd^og Bka66ov 

10 tov jdZ [iByid^ovg^ iv Sh tp ig)B^ilg \)7totCd'BtaL tijv 
/iZ ikd66ova tfjg ix rov xivtQOv xal Tcdkiv SbCxw6l 
tb AB (liyBd^og BXa66ov tov AZ {iByid^ovg, 

79. 'ATtb trig 0N p. 224, 20] iTtBl yctQ iXd66cjv 
iTtBtid^rj tfjg ix tov xivtQOv^ rj Sh &N ix roi) xivtQOv^ 

15 (ibC^cov d^ikBL Bivai tfjg Zjd tr\g iXd66ovog, 

Ad prop. XLI. 

80. ^^g i%\ t&v a6tQG)v. 

81. Tb avtb S\ 6vn^ri6BtaL^ xal bI tb ofi/Lta iTtl 
tov xivtQOv roi) xvxkov ^ivBL^ tb Ss 6q(6iibvov ijtl 

20 tfjg TtBQLcpBQBCag [iBtaPaCvBL. 

82. ''E6tL tLg t^itog^ ov roi) Sfi^arog [iBd^L^ta^ivovj 
tcbv Sb 6QC3iiivG)v t6G)v fiBv6vtG)v xal TtQbg dQd^o^g rc3 



77. V^ 78. V^ 79. V^q. 80. VRVat.FM^ptu. 

81. RVat.M^u. 82. V mg., signo % post prop. 41 (in 

cod. ^f; prop. 42 in cod. iid"' est) insertum; in fine est: ^ijrsi, 
rb d^Sfhgriiicc slg rb 'nocrsvocvrlov; est enim in pag. pr. folii 
sequentis. idem theor. habet q in textu post prop. 43, quae 
in q est v' (ft^' m. 2), numero ^rj' signatum {v' m. 2); ad 
prop. 41 (ftj' q) add. fTfrft ^?]'; prop. 42 est ffS-', fir^' m. 2. — 
Bc re cfr. opt. uet. prop. 46, ubi u. fig. 



IN OPTICORUM RECENSIONEM THEONIS. 279 

"VTCoxsL^EVG) aTtLTtadG) ^ jcoth [ihv t6a^ itorh 8\ &VL6a 
€paCvBtai, 

i^tcD t6a ^syad^iq ta AB^ JTz/ tcqoq dQd^o^g 'ovta t& 
i)7eo7cst^dvG} s%L7CaS(p, kayto^ ort i6ti, tig tdjcog^ ov tov 
iililiatog ^sd^L^taiiavov^ tov dh dQco^avov [lavovtog^ t& 6 
AB^ FA %ot\ iikv t6a^ Tcota 8\ avi6a q^aCvstav, ijcs- 
t^svxd^io ii BA xal tst^T^^d^ix) 8C%a Tcat^ tb E^ xal ilxd-to 
uCQog dQd^ocg aitfj 7] EZ. Xaycj^ ort, iAv iicl trjg EZ 
rb Hiiiia tad'?}^ tcc AB^ FA t6a (paCvatai. xsC^d^co y&Q 
iTcl tov Z, xal 7CQ06jajctatco6av &xtlvsg al BZ^ ZA^ 10 
Zr^ ZA. t6Yi aqa fj BZ tfj ZA. aUct xal i^ AB 
xri FA iTcdxsLtat t6ri' 8vo 8ii al AB^ BZ 8v6l tatg 
JL^, AZ t6ai sC^Cv xal ycovCag dQd^&g %aQia%ov6vv' 
^d^ug aqa ii AZ ^dfiai tfj ZF t6ri i6tCv. &6ts xal 
ycjvCa ii i)7cb BZA tfi i)%b AZT t6rj. &6ts ta AB^ 15 
r^ t6a d(pd^6stai. fistaxsC^d^co 8ii tb Sfifia xal i6tcj 
rb H. kayc3^ ort avL6a f)(p%"^6atai. 7CQ067CL7Ctatc36av 
axtivag at HB^ HA^ ifF, HA. iiaC^cov ccQa i^ BH 
rfjg HA. d^prjQiii^d^a) oiv d7cb tTjg HB tfi HA t6ri 
il BS^ xal i7Ca^avx%(o ij A@. t6rj aQa ycovCa ij i)7cb 20 
BSA tfi i)7cb FHA. dW ij i)7cb BSA tf^g i)7cb AHS 
^sC^cav ixtbg ydQ' xal i] i)7cb FH^ aQa tfjg i)7cb 
BHA fisC^cjv. &6ts xal ij JTz/ fisC^cov tfjg AB (pa- 
v^^staL. 

Ad prop. XLIII. 25 

83. 'Eg)dtl)StaL di^ p. 228, 24] idv ydQ tQstg si^&slaL 
avdkoyov 5)6 lv^ tb i)7cb t&v dxQmv t6ov rp d^cb tfjg 
lii^rjg^ xal Sid tovto 8Ld tb Ad' tov y' tf\g 'E7CL7ca8ov 
i(pd7CtataL. 

83. V. 



280 SCHOLIA 

84. ^'^XXcag ro v\ 

i6T(o bQ&iiLSvov fisysd-og tb K/d^ sifd^sta dh jcXayCa 
S6rc3 ii BF^ xal nQO^SK^s^lri^d^G} iiC si^siag ry jdK 
ij KF ocal 6viLfiakksx(o tfl BF TcatA tb F, xal £iy^q>^io 

5 tcbv ^r, FK iis6r] avdkoyov ii FZ^ xal i6t(o rb ifbfia 
tb Z, xal iista7csxLvrl6d'C3 tb Hfi^a tb Z xal i6t(o inl 
tf^g avtfjg siO^sCag tb B. 

ksyo^ Zrv tb inb r&v -^ 

Z, JB 6Q(ofisvov avi6ov 

10 (pavifi6staL, iics^svxd^cs- 
6av svQ:stal aC KZ^ 
Zjd^ KB^ Bjd^ xal ys- 
yQccg^d^CD tcsqI rb KZ/i 
tQLycjvov tiiYiiLa x^dxXov tb KZ^^ xal xsi^d^cj tf^ vnb 

15 t&v FBJ yGivitx, t6i] fj vjtb t&v FKH^ xal ijts^svx^(o 
rj H^. iv xiixkip ccQa i0rl tb ^KHB. iutsl ovv 
lisi^cov fi {)7cb KZA trjg i)7tb KH^' iitL^svx^si^rig y&Q 
trjg OK (pavsQbv tovto' l'6r} Ss 7] {jjtb KH^ tfl vjtb 
KBjd^ insiSriitsQ iv tS avrp t^TJfiari i6tiv^ xal ij 

20 ijtb KZ^ ccQa trjg inb KB^ fisi^cov i6tiv. 

\ sxoiLSv yccQ' t&v iv totg xvxkoig tstQOJtksvQcov al 

ccTtsvavriov ycoviai 8v0lv dQd^atg i!6av sl6iv* &6ts xal 

tb ccvti6tQO(pov' iccv tstQajtksvQOv aC ditsvavtiov Sv6lv 

dQd^atg l6ai &6lv^ iv xvxX(p i6tl tb tstQdjtXsvQOv^ &g 

25 Ssi%o[isv. iitsl ovv ii vTcb FKH t6r] tf] i)7tb FBjd^ 




84. V mg., q (prop. 43 in V est v')\ cfr. opt. uet. prop. 42 
^XXog. Lin. 21 sq. pertinet ad J KHB lin. 16, quo signo V 
refertur in Vq. 



11. ai] q, m. rec. V. 15. FBJ] e oorr. m. rec. V, 

rjB q. 16. t6] q et corr. ex tc5 m. rec. V. 20. rfjg] q 
et corr. ex rfjt, V. 



IN OPTICORUM RECENSIONEM THEONIS. 281 

xoLvrj 7tQo6xsi0d^(o 7] vjto HK^' aC ijtb FKH aga 
HK^ ratg x)7cb HKJ^ HB^ t6ai. alTC aC vjtb 
FKH^ HK^ 8v6lv dQd^atg i'6ar xal aC x)7tb HK^^ 
HB^ aQa 8v6lv d^d^atg t6av sC6cv. &6ts xal aC koi- 
TtaL or6 da, iav xsxqaitXsvqov aC aitsvavxiov 8v6lv 6 
dgd^atg l'6at &)0lv^ iv xiixXo) s0xl xb xsxQccTtXsvQOv^ 8s- 
8sixxai iv xip iTtofiviliiaxi, 

Ad prop. XLV. 

85. Tb ai)xb rp i/j8'. 

S6Xi xig xditog xoivog^ iv S xov S^fiaxog xsd^ivxog 10 
xa l'6a fisysd"rj avi6a (paCvsxai. 

l6xcj t6a fisyid^ xcc AB^ BF^ xal i^x^^ ^^ '^^ -8> 
TtQbg dQd^&g ^ B^ xal ix^spXif^^d^G). kiyco^ Zxv ^uxX^^A 
bitoiovoyv xrjg Bjd fiiQog xsd-fi xb Sfific^, xA AB^ B J^ 
t6a (paCvsxai, xaC i0xi avxdd^sv 8r}kov. iisxaxs£6d^m : 
8^^ xb Sftfta xal l6x(o xb E. Aeyco, Sxc djtb xov E 
avi6a (paCvsxai. 7tQ067ti7txix(x)6av y^Q axxtvsg aC AE^ 
EB^ EF^ xal ysyQcc^pd^co itSQl xb AFE XQtycjvov xihcXog^ 
xal ix^s^kri^Q^G) fj EB inl xb H. iitsl oifv l'6rj fj AB 
xri Br^ iisi^cov 81 ^i FE xrjg AE^ iisC^cdv &Qa xal 2C 
ycjvCa rj x)7tb xcbv AE^ EB xrjg x)7tb x&v BE^ EP. 
fisC^cov aQa (pavr}6sxai 'fi AB xrjg BF. h^aiixcjg Sd^ 
xav fisv ijtl xrjg BZ T£'9'iJ, C6a (paCvsxat^ iicv d^ i^^ 
xfjg BH^ avi6a. bfioCcog 8s xal iTtl x&v aklcov to^ 
xvxXov [iSQG)v x^Q^S ''^VS ^Qbg dQd^o^g idcv xsd^fj xb SfiiicCy ^ 
I avi6a (paCvsxai. 

85. Vq. (post 46, v^' V) (V in mg. inf.). 



2. av\ comp. V, seq. ras. 3. cti'\ in ras. V. 9. t6\ ^ 
corr. m. rec. V. 17. ocl'\ q, om. V. -<iE] q et corr. ex i -^ 
m. rec. V. 26. tintSL tb -O-eebQTiii.a Siti^tv N . 



282 



SCHOLIA 



86. Xhir di dvptetiyv Tiiiv£6^ai rb ijfuxMUov imb 
tov luC^ovog riirfiucrog xal xov^ ovtag &ftai dfiJLov' 
l6r(o6av t6(u at AB^BF^ xol xsQiyeyQdtp^a f^iuxvxXiov 
Ttigl tb AB tb ASB^ tucI 6we6taxfo xgbg xf^ BF 

d 




6 xal ro B 67iiucm ycovca d^sta ^ imb FBA^ TCQog de 
ro r t67j rfi B 71 r^ otal 6vii7tim£t(o ocara rb ^, ocal 
icTcb rov A iTcl rb xivrQOv rov A&B^ o i6rL rb E^ 
iTCs^svxd^co 7] AE^ xal xst^d^fx) rj5 BZ xsQKpsQsta t67i 
7] Z®^ xal iTtstsvxd^co^av aC AS^ @E. iicsl ovv ^ ®E 
10 r^ EB 1671^ TioiVTi ds ti EA^ xal ycovtag t6ag tcsqi- 
i%ov6iv^ ijcsl xal 7] SZ nsQLtpsQSia rfj ZB i6rcv t67j^ 
t67i aQa fi 0A tfi AB. 7] ds AB rfi AT' &6rs 6 
xivrQO) rp z/, 8La6r7]iiart Si r^ A& yQatpoiisvog xihckog 
rsiisl ro TiiiLxvxkiov xal dta rov B iksv6srac. 

15 Ad prop. XLIX. 

87. !Ex rov d^scoQTjiiarog (pavsQtorsQOv ytvsrac rp 
6vii7cC7crsiv avrd. 

86. Vtt(Vat.Aqru, in textu t). 87. VR(FVat.qt). 

1. dvvcctdv] VVat., dvvcctcci R. 3. at] om. VR. xat] 
om. R. 4. x6 (pr.)] td RVat., et V, sed corr. x6 (alt.)] 

om. VR. 6. ii\ eras. V. 12. c5(?Tf] gxb post lac. Vat. 

13. zfj Bupra scr. Vat. to5 40] R, t oidQ' Vat., r» ydQ" e 
corr. m. rec. V. 



IN OPTICORUM RECENSIONEM THEONIS. 283 
88. 0£QO(iev(ov (bg L7t7CC3v tv%ov &3th xCbv aQL0tS- 



QG)V iicl ta 8£%L&. 



Ad prop» L. 



89. Olov Tckoicjv. 

Ad prop. LI. 

90. ^Slg €7tl toixcov. 



Ad prop. LIII. 

91. T&V t6G) tA%£L g}€Q0ll€VC3V ta TtdQQCO 807C£L 

fiQa8vtEQ0v g^EQe^d^aL. 

g^^Qded-G) yccQ dt5o 6ri^€La't& A^B 10 
iitl TtaQakkifiXGiv Eid-^L&v tcbv A^^ 
BE diiaX&g' td:g l'0ag ccQa iv 1'6g) 
^ Xq6vg) 8L€X€v6€taL, l6tG)6av ovv 
t6aL al Ajd^ BE^ xal 7tQ06- 
7tL7tt€tG)6av axxlv€g ajto tov Z 15 
(iiifiatog at ZA^ ZA^ ZE, iitel 
oiv iXdttG)v i6tlv '^ iTtb AZA 
yG)vCa tr^g x)7tb BZE^ iXattov 
ccQa tb AA 8Ld6trjfia tov BE g)av7^6€taL, &6t€ 86^€l 
tb A ^Qa8vt€Q0v g^iQ^^d^aL tov B. 20 




88. VVat.(q). 89. VRFp. 90. VRF. 91. VR 

(Vat.Mqru, in textu post prop. 63 F et add. numero v^' t). 



1. tnitoiv] titm Vat. 6. xol%(ov'\ xv%(ov V, corr. m. rec. 

8. yLQsLxx(ov ccvxri ^ &7c686L^Lg PR. taca] ho V, sed corr. 

xdxL V. 12. 6iiaX&s] om. R. 17. AZJ] V, ZA M. ^^ 

postea add. J B>. 19. &Qa — 20. B^ om. "^. 



284 SCHOLIA IN OPTICORUM RECENSIONEM THEONIS. 



Ad prop. LIV. 

92. "E6tG} 6Q(o^eva rcc A^ T iitX 7CaQakX7]ka)v ^ta 
t&v AB^ jTz/ sid^Bimv, Xiyo^ Sti tb xdQQCs tb A 
xatalsLTtead^aL S6^sl. i6t(o yag 
6 a^^a ro E^ dq)' ov 7CQ06%i7ttst(o- 
6av dxttvsg al EF^ EA^ EJ^ 
EB, ijtsl ovv liSi^cov i^tlv fj 
x)7cb FE^ trjg x)7cb AEB^ fist^ov 
aga xal tov AB tb F^ (pavT]- 
10 0staL. iTCoXsCTCstai aQa ro A' 
Soxst yaQ ^qaSvtSQOv (piQS^d^ai, 




92. VR(Vat.qrtu). 



2. inl 7taQa%XriiX(ov] iit s^bd^siccg V. 
svd-SL&v Vat. 9. tov] i} rd, 17 eras., V. 
Post A eras. J V. 



3. svd^SL&v] faav 
rd] tov V. 10. 



CATOPTRICA. 



'OtpLV alvai svd^stav^ 'Ijg xk iis6a Tcdvta totg aocQOcg 

Tdc dQih^sva aTCavta Tta^"! svd^siag bQafSd^ai. 

'EvdmQOv tsd^svtog iv mi^iSfp xal ^sc^QOVfievov 
6 tLvbg iiil^ovg^ b TtQbg dQd^dg i6tv t& ijcmidp^ yCyvovtai 
avdXoyov^ hg fj fista^i) tov iv6%tQ0v xal rov ^sm- 
Qovvtog sifd^sta TCQbg tijv fista^v tov ivdictQOv xal rov 
^Qbg dQd^&g iiipovg^ oiitG) tb tov d^sioQOVvtog v^og 
jCQbg tb TCQbg dQd^o^g tp iTCLTCsSG) iitl^og, 
10 ^Ev totg ijCLTCsdotg ivdjctQOig tov tdTCov xataXrjg)' 
Q^ivtog^ iq)^ hv rj xdd^stog TCiictsc dicb tov 6q(oii8vov^ 
ovxitL bQatai tb oQafisvov. 

Kal iv totg xvQtotg ivdictQOig xataXrjg^d-ivtog tov 
toTCov^ 8l^ 01} dnb tov 6q(0[isvov slg tb xsvxqov aystaL 
15 trjg 6(paLQag^ oixitL oQcctaL tb 6q(x)^svov. tb d' avtb 
xal iv totg xoCkoLg 6vii^aCvsL. 

'Edv slg dyystov iit^lrjd^Ti tL xal Id^ri d7c66trjfm d)g 
firjxitL 6Qa6&aL^ tov aitov djco^f/nnatog '6vtog idv 
iidcoQ iyxvd-fj^ h^pQ^ifi^staL tb iiipirjd^iv. 

20 a\ 

^Ajcb tcbv iTCLTcidcov iv67CtQcov xal xvQtcbv xal xoC- 
Xcov al oipSLg iv l'6aLg ycovCaLg dvaxkcbvtaL. 



"Oqoi m, Zqol xaroTtTQtTtdav m. rec. v. 1. Supra s^b^slav 
^g 8cr. {fTtOTiBiGQ^Gi m. 2 \ , mg. m. "V'. -«.axw. -«.^ivv^^i x>i W^- 
xsia&o). fig'\ corr. ex slg v. ^. icxwN^. tv-v^-vxo.v ^. 



Uisum rectam esse, cuius partes mediae omnes 
extremis officiant. 

Omnia, quae cemantur, secundum rectas cemi. 

Ubi speculo in plano posito altitudo aKqua ad 
planum perpendicularis spectatur, proportionem habet^ 
ut recta inter speculum et spectantem ducta ad rectam 
inter speculum et altitudinem perpendicularem ductam^ 
ita altitudo spectantis ad altitudinem ad planum per- 
pendicularem. 

In speculis planis eo loco occupato, in quem recta 
ab eo, quod cemitur, perpendicularis cadit, illud non 
iam cernitur. 

Etiam in specuKs conuexis eo loco occupato, per 
quem recta ab eo, quod cemitur, ad centram sphaerae 
ducitur, illud non iam cemitur. idem autem etiam in 
speculis concauis euenit. 

Si res aliqua in uas coniecta et tam longe remota 
erit^ ut non iam cematur^ si eadem distantia manente 
aqua infasa erit, res in uas coniecta cemetur. 

1. 

A speculis uel planis uel conuexis uel concauis 
radii sub angulis aequalibus refringuntur. 

6. tov (alt.)] m. rec. V. 8. ovta)] ovro} nai M. ^t(»Offl ■ 

corr. ex v^ipovg m. 2 v. 13. roaj^ e coTt. tel. \X>» \i.^x«xv 
dp&Ts F, sed corr. 19. ^yxv^fl^ iY.XS%Tft ^-^ ., '^H»'^ ^ 'a^ "" 
rec; ^rz^^tl ^9 &yyBlm m. ^O. a'^ om. '^. 



^^i catoptrica. 

iaziD uttuc ro B, sTfyxTgnT htijisdam ro ^I\ o€*ig 
d' cxo xoi' o«iu:toc ^s^d^&a i; BK sei awwauxlao^ 
ixl ro _/. ^roi «Jij; r^r £ yovienr i&f^ iivtu rg Z 
r^l^Qi6€LV zil&srot ixi ro ^roxrpor cu H/^ ^^. oi^- 
5 ow i^ir, ci^ r^ BF :rpo^ FJu ^ ^^ ^pi^s ^K' Totrro 
vc{> iv ror? o{M>cc vxixiiTO oaoiav Sga rb BTK tqi- 
yarovTo ^AK TQiyorw. i6r^ cgc r^ E y&via rg Z 
ym'Ca' rc ytLQ ouoia TQiytJva iMy&via i&nv. 

i6T& dr^ an'pT«>r iv- 
I0'ocrrpor rb AKF^ ovtg de 

rb A. jiJy(3, &ri i6r^ i6Tiv 

Ti E^ S yavta rg Z, ^i. 

xagi^^Tca aixidov iv- 

15 orrrpor rb AW' ftfij c|m: 

/tfrir 15 £r y&via rjj Z. CAJlc xc2 ij ^ ^S ^" «9?- 
crrrfra* ^cp i; 3/A'. oaij apc i; £1» © oilj Tg A^ Z 
^'«yrti/ r(5i;. 

^«^ro djjj rraAtr xofAor ivoxrgov rb AKF^ oi?ig ds 

20 ff BK uvaxhoaivr^ ixl rb A. ^yo^ ori ^ E ycrvia 

i6r^ 16x1 rg Z. Tcagaxa^ivxog yag ixiTcidov avoxTQOv 

i6r^ yiyvtrai ii B^ E y&via rf^ Z^ A' i6ri Sh xal ij B 

rfi A' koiTcii aga ff E rfi Z i6r^ i6rai. 




1. Post JB add. tcuL m. rec. V. 2. BK] BE M. 6. Ante 
rX add. trjv M, m. rec. V. AK] tfjv AK. AK e corr., M; 
ri}v add. m. rec. V. 6. orrfxfiro] mut. in i:t6%siTat m. rec. V. 
7. Post 5^a add. iGtlv m. rec. V. 8. tQiyava] om. M. 9. 
p' V. 10. ^Xr] corr. ex AK m. rec. V. 15. rb NM — 
18. Ttfij] eras. V, m. rec: tb NM xal i:tel iari iatlv ij w(0 
MKB ymviat^ ynb NK^, aXXa yud 7} v^b FMK rg {ntb AKN' 
i<pdntttai yuQ 7} MN' oXt] uQa i) V7tb BKF ty vn6 ^KA foii 



CATOPTRICA. 



289 




sit oculus Bj speculum planum AF^ radius autem 
ab oculo feratur JJjfiT et ad ^ refringatur. dico, 

esse /. jB = Z. ducantur 
ad speculum perpendicu- 
lares BF^ JA. erit igitur 

BT.TK^ JA\AK', 
hoc enim in definitionibus 
suppositum erat. itaque 
trianguK BTK, JAK si- 
miles sunt. qaare erit i E = Z] similes enim tri- 
anguli aequianguli sunt. 

iam conuexum sit speculum AKT, radius autem 
BK did A refractus. dico, esse iE-{-@=Z'\'jf 
adposui speculum planum NM, itaque i E = 1 
uerum etiam iS = A\ MN enim contingit. erg- 

iE+@=^A+Z, 

iam rursus concauum speculum sit AKT^ radios 

autem BK a.d A re&actus. 

dico, esse iE=Z. adposito 

enim speculo plano fit 

i® + E=Z + A. 

uerum etiam i@=A. ergo, 

qui relinquitur, i E = Z. 




iativ. 17. E, @] @,E in. ras. M. 19. y' Vv. 20. BJH 
BE M. E ycavia] mut. in vn6 BKV yiovia m. rec. V. 

21. i6tiv V. Z] mut. in vith JKA m. rec. V. 22. O, JB] 
mut. in vnb BKM m. rec. V. Z, A] 'bnb JKN m, m. rec. V. 
xal 7} t^ A] 7) vTtb FKM rj i)nb AKN %cci m, m. rec. V. 

23. 7j E — Uxcti] ij {jTtb BKV rj vitb J KA tisri imlv m, 
m. rec. V. 



Enolidds, edd. Heiberg et Menge. 'VIL. 



Vi 



■^ 



290 CATOPTEICA. 

IlQbg bnolov av t&v ivdxtQmv TCQOtfxdfSri a^tg tfSag 
noiov6a ycaviag^ axnii 8i iavrilg ivaxXaff^^ffsrcc^, 
i6t(D ivoTCtQOv ijCLTceSov tb AF^ HfLfia dh tb 5, 

r> (iil^tg di 7^ BK 7CQo67CS7Ct(Dxit(o t6ag xocavtfa ycyviag 
tiiv jB, Z tfj @. A^yco, otc dvaTcXcjfidvr^ i^ BK i(p 
iavtf^g ^^Bi^ tovti6tiv iicl tb B. fiij V^Q^ dXK^ ei 
dvvatov^ '^xitco iicl ro ^. xal iiceLSij a[ fi^sig iv t6aig 
dvaxKbvtai yoiviaig^ t6rj i6tlv fj E ytovia r§ ft 

idtix&rj Sl xal fi E^ Z ycjvia tfj & t6rj. xal ^ E, Z 
(iQ(i yc3via tfj E yG)via i6taL t6rj^ i^ M^^'^ '^H i^<f- 
tfoi/f oTCBQ i6tlv dSvvatov, ij aQa BK 8i aircrjg dva- 
xka6d^68taL. ii d' aitij d7c6Ssi^ig dQ[i66ec£v av ari 
ru)v xvQtmv xal tcbv xoiXav iv6%tQ(ov, 

f* Y' 

JlQog 67COLOV ctv tcjv iv67CtQC3V 7CQ067ci7crov6a StjfLg 
(ivi6ovg TCOLfi ycoviag^ ovts Sl savtrig dvaxka6d^6stou 
(tvtf- iTcl trjg iXcc66ovog ycoviag. 

i6tc3 iTCLTCsSov svoTCtQov tb AKF^ Stlfcg dh ij BK 

7tQ(i67CL7Ctsto [isi^ova 7C0L0v6a yoviav tijv Z rfig 0, A, 

kiyu)^ otL rj BK dvaxkcoiiivrj oiits avtij dt' iavtilg 

dvuxXa6%"ifi6staL ovts iTcl trjv @y A ycoviav. si ^v 

\. p'\ d' Vv. 2. TtQoajteaoL M. Dein add. ij m, m. 

rur.. V. G. f^v — &] tag vnb AKB, TKB in, m. rec. V. 

HK\ BE M. 8. TjTieTa)] hsra) M. oipLg v, corr. m. 2. 

II. iSJ vnb AKJ m, m. rec. V. @] ^nb VKB m, m. rec. V. 

10, E, Z (pr.) — &] V7tb AKB tjj vicb FKB m, m. rec. V. 

Z^t.)J inb AKB m, m. rec. V. 11. E] ^jr6 AKJ m, 

ymvicc ^aTat] iaxiv m, m. rec. V. iXdzxovi M. 

Om. M. BK] BE1&. 8v a^r^s] 6ipig iq>' 

lec. V. avtiig] mut. in kavtfjg m. 2 t. 13. 

fsis %al m, m. rec. V, &v\ M, om. Vmv. 





CATOPTRICA. 291 

2. 

Ad qualecunque speculum radius adciderit aequales 
efficiens angulos, secundum se ipsum refringetur. 

sit AF speculum planum, oculus autem B, radius 

uero JB K adcidat aequales angulos efficiens E-}- Z=^@. 

dico, BK refractum per se ipsum uenturum esse, h. e. 

A K ^T ad 5. ne ueniat enim, sed, 

si fieri potest^ ad /1 ueniat. 
et quomam radii sub anguUs 
aequalibus refringuntur [prop.l], 
erit /. jB = 0. uerum etiam 
/. E + Z = 0. quare etiam 
/. J5 + Z = E, maior minori; quod fieri non potesi», 
ergo BK secundum se ipsum refringetur. eadem' 
autem demonstratio in speculis conuexis concauisque 
conueniet. 

3. 
Ad qualecunque speculum radius adcidens inaequa- 
les angulos effecerit, neque secundum se ipsum re- 

fringetur neque ad mino- 
rem angulum uersus. 

sit planum speculum 
AKTy radius autem BK, 
adcidat efficiens 

/. Z > ® + ^. 
dico, BK refractum neque 
secundum se ipsum neque ad angulum + ^ uersujB 




16. y'] 8' Vv. 17. noi'^ noist M, et m, sed corr. IB^ 

BK] BE M. 20. Z] vnb AKB m, m. rec. V. O, A] *«fA 
rKB m, m. rec. V. 21. BK] BE M. 22. ti^ B^At 
vlotv] xfig ijTcb BKF yoiviag m, m. twt. "NT. 



292 CATOPTRICA. 

yaQ ^^sc iitX to B^ i6tai ii Z yiovCa rg ®, A USr^' 
OTCBQ atonov' vTcdxsctai y&Q iis^cav. sl dh 9i£t tov J^ 
t0ri i6tav fj Z ymvla tfj 0' i6tv Sh ftftgcDt^. ij &Qa 
BK dvaxka6d^6stai ijcl tiiv fui^ova yayvCccv rijv Z* 
6 Svvatov y^Q oato tflg [isi^ovog tfj ika66ovv t6riv dq>atQS- 
di^vat. i6ti 8\ fj aiftii SatdSsc^ig ixl tSrv tcoqx&v xal 
KoCkGyv. 

At ^tjjSLg iicl t&v imTciSiov iv6ntQ(ov xal xvQt&v 
10 dvaxXcoiisvac ovts 6v{i7Cs6ovvtac aXM^Xaig ovxb tcoq- 
&kXYikoi i6ovtai. 

i6tc3 iTCLTCsSov ivoTCtQOv tb AF^ ififia dh tb B, 
HxlfSLg Sh avaxXfhiiLSvaL at BFA^ BAE. kiyG)^ 5tc at 
FA^ AE oiits TcaQalXrjXoi s16lv ovts 6v[i7CS6ovvrac izl 
16 tcc A, E. iTCsl yaQ t6r} i6tlv ^ Z yayvCa tfj ®, '^ Ss K 
tf] M^ fisL^c3V SsiiZ tfjg K Slcc tb ixtbg slvai iv 
rc3 BAF tQiyiovG)^ fisitcov ctv stri xal fi & tfjg M, oix 
aQa TcaQakXriXog ij FA tfj AE i6tiv^ o^bS^k 6vfi7CC7Ctov- 
6lv iTcl ta E^ A. 

20 i6tG) TcdXLV xvQtbv ivoTCtQOv tb AZT^ SfJL(jLa Si 
tb J5, 6tl^sLg Ss avaxXcjiisvaL at BZA^ BHE. XsyOn 
otL at Zz/, EH ovts TcaQctXXrjXoL s16lv ovts ^vfi- 

1. B, EataL] JB A V m. 1, ^ htaL m, m. rec. V; BE e 

I 
corr. M, BX V. Z] vith AKB m, m. rec. V. 0, A] 'b^b 

FKB m, m. rec. V. 2. sl 8b — 3. fig/Jcoi;] om. M. 3. 

^(yrt] ^(XTn; Vv. 4. BK] BE M. r?iv \LBiiova — Z] rijg 

lisiSovog yaviag tijg vith AKB m, m. rec. V. 5. tariv] Hgov v, 

et V, corr. m. rec. 6. feti; Vv. 8. d'] g' v et in ras. V. 

15. Zl ft^i; vTib BFZ m, m. rec. V. 0] 'b'Jib dPA m, 

m. rec. V. K] vnb BAF m, m. rec. V. 16. M] M 

EAH m, m. rec. V. ^ei^tov] e corr. v. Z] vnb BFZ m, 

m. rec. V. K] vitb BilF m, m. t^c. V. iv t<5] rov m, 



CATOPTRICA. 



293 



refractum iri. nam si ad B uenerit, erit /, Z = ® + ^; 
quod fieri non potest; supposuimus enim Z>@ + ^. 
sin per ^ uenerit, erit /. Z = 0; est autem Z > 0. 
ergo BK a.d angulum maiorem Z uersus refringetur; 
fieri enim potest^ ut a maiore angulus minori aequalis 
auferatur. eadem autem demonstratio in eonuexis 
coneauisque ualet. 

4. 
Radii in speculis planis conuexisque refracti neque 
inter se concurrent neque paralleli erunt. 

sit planum speculum 
AF, oculus autem JB, 
radii autem refractiB J!^ 
BAE. dico, r^, Al 
neque parallelos esse nfr 
que concurrere ad z/^ £ 
uersus. nam quoniam 
iZ = S, LK = M, uerum /. Z > jRT, quia in tri- 
angulo BAF extrinsecus positus est, erit etiam 
L S> M, ergo FA neque rectae AE parallela est, 
nec concurrent ad jB, ^ uersus. 

rursus conuexum sit speculum AZF, ocuIub 
autem B, radii autem refracti BZA, BHE, dieo 
ZAj EH neque parallelos esse neque ad E, J uersi:i.i^ 

m. rec. V. 17. JB^T] m. rec. V, ^KV v, m. 1 V. r^e- 

yiivov m, m. rec. V. ^v\ &qoc M. ccv sHti^ &Qa iavi 
m. rec. V. 0] {)n6 J FA m, 'bTtb JFA m. rec. V. 
vnb EAH m, 'bnb ISlAH m. rec. V. 19. Post ^ add. o: 
&lXrilaig m, m. rec. V. 20. J' Vv. Post lcrrcB add. ^^ 
m. rec. V. AZF] M, AlSir v, m. 1 V; AHT m, m. rec. 

21. BZz^] Z add. m. rec. v; BdZ M. BHK] H 

m. 2 V. 




294 CATOPTRICA. 

X£6ovvrat inl xa E^ jd. ixBlsvxQ-m yhg 1} HX ei^la 
xal ixPefiXii^d^ai iq>^ ixatBQo, ixBl t6fi iaxlv 1} JT , 9 
T§ A dik t6 iv t6cug ivaxXatfd^ai yovuugj ehi aw 
luCtjayv fi A^M T^g K. fi Sk K rf^s N, S iffri (lei^j 
b fi dl N^ S rrjg O, 77 luC^ayv' ainii yig fj S t6fj M 
ty O, 77* (uC^c^v aQa fi A^ M xf^g O, 77. xoXk^ &f€ 
il A^ M xf^g O ^sC^mv i6xiv. oix afa avfug£6ovvxai 
al ZA^ HE svd^etat oidl jcaQdXXrfXoL sUtiv. 






^Ev xotg TcoCkotg ivonxQOig iav ^ inl xo xbvxqov 
^ inl xfjg XBQitpBQsCag ^ ixxog xf^g XBQifpBQsCag d^g xh 
o[iiuCy xovxi6xi luxal^v xov xivxQOv xal x^g XBQupBQBCag^ 
cd ^sig ivaxkioiisvcu 6v[ixs6oi}vxai. • 

s6XG} xolkov ivonxQov tb AFA^ xsvxqov d% r^ 

5 6tpaCQag tb B^ xal xsC^d^fo tb oftfia ixl xov B^ xal 
nQ06XLXtstc36av ccTtb tov B otl^SLg XQbg ti^v xsQLtpsQSLav 
ai BA^ -BF, BA, t6aL ccQa si6lv cci XQbg totg 6r^- 
[isCoLg totg A^ z/, F ycovCaL' f}fiLxvxXCov yccQ bl6lv. al 
&Qa otl;sLg dvaxkcjcLSvaL Sl^ savtd)v dvaxla6diJ6ovtcu 

!0 aC BA^ BF^ BA' tovto yaQ SsSsLXtaL. &6tB 6vfi- 
xs6ovvtaL xatd ro B. 

i6ta} xdkLV xotkov svoxtQov tb A FB^ 5ii[uc dk xb B^ 



1. HZ] Z M. 2. Post iytdtsga add. xara ra ©, K tfij- 
fula %ai m, m. rec. V. Post tar] ras. 1 litt. V. K — 3. ,4] 




(LSl- 

« V, corr. m. 2. 8. ZJ] JZ m. 9. «'] ij' Vv. 10. 
Qov] tov xsvTQOv m, m. rec. V. 11. b^fjg] ^Btg V, 




CATOPTRICA. 



. 295 




concurrere. ducatur enim recta HZ et in utramque 
partem producatur. quoniam i K -{- @ = A, quia 

radius sub angulis aequaU- 
bus refringitur, erit 

LA + M>K. 
est autem 

LK>N+S, 

N+s>o + n', 

nam S "= O + 11, itaque 
A + M>0 + n. multo 
igitur magis A + M> O. 
ergo rectae Zz/, HE neque concurrent neque parallelae 
sunt. 

5. 
In speculis concauis si oculum in centro uel i| 
ambitu uel extra ambitum, h. e. inter centrum ei 
ambitum, coUocaueris, radii refracti concurrent. 

sit speculum concauum ATid^ 

centrum autem sphaerae B, et 

in B oculus ponatur, adcidant^ 

que a £ ad ambitum radii ^A^ 

B r, B^, anguli igitur ad 

A, jA^ r puncta positi aequaleB 

sunt; semicirculi enim sunt. itaque radii refracti 

secundum se ipsos refringentur BA, BF, B^; hoc 

enim demonstratum est [prop. 2]. ergo in B concurrent. 

rursus speculum concauum sit AFBy oculus autem 




corr. m. rec. 12. tovtsattv Vv. 14. AFJ] ABF m^ m, 

rec. V. tr^g etpalQag] om. m, del. m. rec. V. 17. al(pr.) — 

BJ] om. M. 22. ^' Vv. Post ^ato) add. ^i} m. S 

ArB]ABrM. 






x^^c^i; ^-: -x. rrj Ti^^^.vs^a*^ «cl lvC. zkL cso tdt B 

Ko^z r j K uA-r^r. '^ cjQ<z Z, H xow ©, K tuiiov$ 
i\6\v. i/ff.jcr tjfyfji r A xf2 3/ ijLca&aw* xodJua uoam 
^/^/. rf^ .V. ^vuxz^ovrrtu cpc «u rL/. A£ WKta rb 5^ 
ouo'.o/ df.iiifrfiiTai» tulv ixro: rf^g x^pi^^^iag »2rij 






/vV rots y.oiloig ivdxrgoig iav iva giitfov rov xiv- 

XQoiJ vjiX r^;* :teQi<fiQHag ^f^g rb oiiiuc^ &ri nhv tfvp 

niffovvruL al o^iig dvax?,(0}ievai^ orl de ov 6viJL3ta6ovvxau 

fdrw} HvojtrQov koIIov xo AF^ yJvxQov dh ccvrov 

\r, To •'/, liiiiia dl xHiJd^o x'o B [lexa^v xov xdvrqov Tud 
ry^i^ TTi QUfjf Qf.Lug^ oiptvg dl aC BA^ BF avaxXf&fievai, ixl 
rO li^ Z^ xal ixPi^^pXr}6d-(o6av a[ fiil;eig eiog rov iv- 
uTrTQov (d //W, FK. 7] A& Si] xfig FK 7\ fiet^ov i6xlv 
JJ /Vr»y i) ik(i(S(Sii)v, d ^lv ovv t6ri i6xlv ii AS S^tg 

a<> T/} \'K o(/'f/, iihi iilrl xal yj AF® TCeQKpiQeva xfi r®K 
TTi Qnyi Q('i\x, Knlrt xal r/ M y(ovCa x^ tS' aC yccQ x&v 
iOu)i' TnQtq^tQi^tMV yioviai i'6ai el6\v aXXn^laLg, xal al 
M^ ./ ;'u»rna ixQa ralg /V, S el6LV t6aL Slcc xijv ava- 

1. AV I om, M. l\. Tost. i:(H add. ovv m, m. rec. V. 

^iMJTow' V. Poiiulo juld. fVnr m, m. ree. V. EP rnrjiuc- 
ro»^] N«'HAorfioru^' M. 1. Tost tifi^iov add. ioriv m. 2 m. xal 
— ft. ^fiir«kU'| «Vuv r*; urru t^»; na) t) KII ^eorr. in K ri)j H) 
W 4mv m. l. H\ mut. in K m. ree. supra scr. dia t6 
V; H *W r»;»: *\ Mv, 5. a^>a j»r.'^ del. m, rec. V. 

] mwt. in H m. iw. V. rost u^i^av add. ^<rri m, 

Z* H\ / K M. 01 \\ oerr. m. 1: ZA^ m, m. rec, V. 
M. K ^ * luut. iu ^>, :{ K T^o. V. 6. sid M. 




CATOPTRICA. 



297 




By ponatur autem in ambitu eius, ei o, B radii ad- 
cidant BFj BA ad puncta J, E refracti. quoniam 

segmentum AFB segmento 
B r maius est, erit /, Z > 0. 
quare etiam H> K [prop. 1]. 
itaque Z + H > S + K. 
quare , qui relinquitur, 
L A < M. multo igitur 
magis A <,N. ergo F^, 
_ AE in. S concurrent. simi- 

liter demonstrabitur etiam, 
si extra ambitum ceciderit oculus, ut in propositione 
sequenti. 

6. 
In speculis concauis si inter centrum et ambitam 
oculum collocaueris, radii refracti tum concurrent, tum 
non concurrent. 

sit speculum concauum AF, centrum autem eius ^, 
et oculus B inter centrum et ambitum coUocetur, radii 
autem sint BA, -BJT ad H, Z refracti, et radii ad 
speculum producantur A&y FK. itaque A@ aut maior 
est quam FK aut aequalis aut minor. iam BiA® = FK^ 
erit etiam arc. AFS = r@K, quare etiam LM=lSt] 
anguli enim arcuum aequalium inter se aequales sunt. 
quare etiam L M-}- A = N-^- S propter refractionem 

Post iXdaaoDv add. iati m, m. rec. V. 7. rfjg N] ii ji 

tfjg N iXdeGODv iatlv m, m. rec. V. 8. Post oiioUog add. 9i m, 
m. rcc. V. TtL^tst VM, et vm, sed corr. 10. s'] *' Vt. 

11. iisGov] iLiaov M. 17. H, Z] N, Z v; Z, if M. 8»g\ 
supra scr. M. 18. Srj] om. M. 19. iXdttmv M. 20. imlif 
Vv. AF©'] AA@ M. r@K] e corr. v. 22. «mm- 

€pSQSi&v — 23. Haai] yfovUav nsQupiQSuci taZg H, I8t taat $U» mg 

23. N] e corr. v. ^ 



208 CATOPTRICA. 

xXa6iv. xal komri &Qa ii O x^ 11 t6fi itfriv. lui^fov 
&Qa ^ P xf^g O. ixsl yoQ r^ P yanfCa xr^g II luC^m 
i6xl dia xb ixxbg elvai^ ii 6\ 11 x^ O l6ri^ ocal ^ P 
&Qa xflg O fieifynv i6xiv. ocoiv^ XQO^xei^d^c} i^ rmb 
5 OPZ. 6vfiX£6ovvxaL &Qa €ct FZ^ AH itg ixl xa H^ Z. 
xb d' avxb i6xai^ xav iisi^aiv r^ AS Si^cg xfjg FK' 
^uitftveg yicQ i6ovxac al A^ M yayviai x&v N^ S^ ^ 
d\ n xf^g O iLBitfiov i6xai xal ^ P xfjg O. i&v ds ij 
AS sv^Bta ikd66Gyv ^ xf^g FK^ dia X& aina ^i^civ 

10 i6xac ^ O yc3via xf^g 11, i6xc di xal ij P xfjg II 
(ui^ov, ovdav &Qa xoXvec i6rjfv eivai xijfv P xfj O t^ 
ika66ova xfjg O, Tcal (lij 6vii7tixxecv xi[v AH xfj FZ. 
tpaveQbv di^ oxc^ xav xe lUcXcov ^ i^ A0 neQiipiQeca 
xfjg FKj iav xe t6rjj ii 6viucx(o6cg x&v avaxXd6eanf 

15 ovxe inl xfjg neQccpeQeiag xov xvxkov ovxe ixxbg ov 
fti) yivTjxac^ dkV ivtbg fidvov. 

r 

Ta vfjjri xal xk pdd-rj djtb xav ejtLTcidov ivojtXQOv 
dve6tQa^li8va q^aivetai. 

20 e6tG) vipog [lev tb AE^ ivontQOv 8i iTtiitedov tb 
AA^ S^^a 81 tb B^ oil;ecg 8i ai BJT, BA dvaxkcjfuvac 
iTtl td jEJ, K. ovxovv cpaivetai ixpkrjd-et^&v x&v oilfecov 
iit^ ev^eiag tb ^sv E tb avca inl tov @ xdxG) Stnog^ 
ro 81 K xdtco bv ijtl tov Z roi> avco ovtog. &6te 

26 dve6tQa^[iiva i6tl tri g)avta6ia. 

1. ieti Mm. 3. iariv Vv. 4. ieri Mm, comp. v. 5. 
OPZ] POZ M. Deinde add. otioiag r& jtgb rovrov ^s&QTJiuxri 
ScTCodsiyLVvrai Vm. ai] al aga M. 6. ^arai] iari M. 8. 
hrai] iari M. 9. FK] FJ M. 10. ^arai] iariv M. ^ffri] 
hnv Vv. 12. iXdrrova M. AH] AK M. 17. f] ta' Vv. 

22. (paivsrai] om. m. 23. r6 (pr.)] (paivsrai ro m. 24. 
Hv] ov rov O m, m. rec. V. rov (alt.)] del. m. rec. V, om. m. 

ovrog] 6vrog rof) m, m. rec. V. 26. iariv Vv, sIgI m. 



CATOPTRICA. 



299 



fprop. 1]. itaque etiam^ qui relinquitur, L O = 11. 
quare L P^ O (nam quoniam £ P > 77, quippe qui 
extrinsecus positus sit, et L n = Oy erit etiam 

LP> O). communis adiicia- 
tur L OPZ, ergo TZ, AH 
ad 77, Z uersus concurrent. 
idem autem fiet, etiam si 

A® > FK', 
nam L ^ + M> N+ S et 
Ln>0, P>0. sin 

A&<rK, 
eadem de causa erit L 0> 11, 
uerum LP> n. itaque nihil 
obstat, quo minus sit L P= O uel P< O, ita ut ^77, FZ 
non concurrant. manifestum est autem, siue arcus 
A& arcu FK maior sit siue aequalis, punctum, ubi 
radii refracti concurrant, neque in ambitu circuli 
neque extra eum fore, sed intra tantum. 




7. 

Altitudines et profunditates in speculis planis sur- 

sum deorsum uersae adparent. 

sit altitudo AE, speculum 
autem planum AA, oculus autem 
B, et radii BF, BA ad E, K 
refracti. itaque radiis in di- 
rectum productis E punctum 
superius in ® adparet inferiore, 
K autem inferius in Z superiore. 
quare sursum deorsum uersae uidentur. 




300 



CATOPTRICA. 



i6to niUv pdd^og fihv tb EA^ IvoTttgov dh isci' 
nadov th AF^ bfifia Si tb ^, 
titlfSLg dl at JF^ JB avaxXc)- 
[isvaL inl ta E^ Z. biLoC&g t&v 
6 8^60)1/ ixpXrjd^SL^&v inl ta 0, jRT 
tpavsttai tb fihv E xdt<D bv inl 
tov & &v(o hvtog^ tb d\ Z avfo 
hv inl tov K ocdtco Svtog. 




10 Td v^Yi xal td fid^rj &nb t&v ocvQt&v ivdntqmv 

dvs6tQaii(idva fpaivstaL, 

i6tG) iifog tb AE^ ivontQOv Sl xvQtbv tb AJF^ 

'dfsig dl at BA^ BF dvaxX6(isvai inl td E^ &. di- 

SsixtaL^ ZtL ov 6v(ins6ovvtaL. td Sl koind 6[ioLG}g 

15 totg iv totg imniSoLg. 

i6r(o ndkiv pd^og tb AE^ ivontQOV 8\ xvQtbv th 

AF^ (i(i(ia Ss tb B^ otlfSLg Sh dvaxX6(isvaL inl xd E^ 8 

aC BFE^ B/IS. td Ss XoLnd xad^dnsQ iv totg inL- 

niSoLg, 
20 ^'. 

Td nkdyLa (iTJxrj dnb tcbv inLniScov ivdntQcov^ &g 
tfj dlrjd^SLcc s^SL^ ovt(o xal tpaCvstaL. 

1. tj3' Vv. EA\ AE m. 2. ds xb J] om. m. 4. 

Ante oiiolcog add. oi^tiovv m. rec. V. d^Loias — 5. ^x^Xfj^a- 
c&v\ oif-Kovv inpXrid^sie&v diioCcog r&v ^i^eodv iit s^b^sLccs n^ 
6. Ante ijtl add. iTc' e^h&siag m. rec. V. Q. 5v] corr. ex 

iov m. 2 V. 7. avco] &vd? M. 8. 6v] ov rov £ m, m. 

rec. V. Post Hvtog add. rov G. xa aga vi/jtj xal tic pdd^i 

StTfb x&v iitLJtidoiv iv67txQ(ov avsGXQainLBva cpalvstai m. 9. 
17 'J i/ Yv. 1-2. AE] A@ Mm. 13. BJ] in ras. V, BF m. 



CATOPTRICA. 



301 



rursus profiinditas sit EA^ speculum autem pla- 
num ^F, oculus autem jd, et radii /dr, ^B ad E^ Z 
refracti. similiter radiis ad 0, K productis E punctum 
inferius in S superiore adparebit, Z autem superius 
in K inferiore. 

8. 
Altitudines et profunditates in specuKs conuexis 

^B sursum deorsum uersae ad- 
parent. 

sit altitudo AE^ speculum 
autem conuexum AAT^ radii 
autem Bjd^ BF ad E, @ re- 
fracti. demonstratum est^ eos 
non concurrere [prop. 4]. 
reliqua autem ut in planis. 

rursus profunditas sit ABy 
speculum autem conuexum 
AFy oculus autem B, et 
radii ai E, & refracti BFEy 
BA&, reliqua autem ut in 
^B planis. 





9. 
Longitudines obliquae in speculis planis, sicut re 
uera se habent, ita adparent. 

JB r] BJ m. 14. oTt] dri Zxi m, m. rec. V. 16. X0I9 (pr.)] 
om. My. Post iitntiSois add. ScjtoSsSstyiisvotg m, ivdxvQo^s 
icjtoSsSBiyiLBvoig m. rec. V. 16. v8' Vv. AE'] A M, AQ m. 

18. BJS] B corr. ex J y. tdc — iMJtidoig'] %ccl ii iatd- 
dsi^ig TtQo^riGsxav diiolcog toTg iv tolg ijtmi^otg &7eodsd$if' 
iiivoLg m. Post miTtsdoig add. ivdTttQoig m. rec. Y. M 
-a-'] t«' Vv. 22. tfj] 7) M. • .^j 



302 



CATOPTRICA. 



OTCTQOV d^ r6 AF. ovxoih/ dva- 
xkaad^SL^&v r&v iixlf6(ov tpaCvetai 
tb [ihv jd hil th A^ tb dl E 
5 ijtl tb r^ %al i6tiv ovt(o tfj 
(pavta6ia^ xad^aneQ xal tfj &kri- 
%da ixu^ tb iyyiov iyyiov^ tb 

ajCibtSQOV OJCGitBQOV, 




10 T& Tckdyia fmlxri &7cb t&v xvQt&v ivdxtQcov^ xa^- 
ciTCeQ i6tlv aXrjd^&g^ xal tpaLVStai. 

i6t€o ^ijxog tb ^z/, Sftfui^ dl tb B^ ivontQov 8\ 
xvQtbv tb AF^ '6tl)Big 8\ dvaxXd)(iBvai ixl tSt E^ A, 
td 8i aXka td avtd. 



16 



ia . 



Td vil;rj xal td pdd^rj dnb tobv xoiXtov ivdxtQcov^ 
86a iLBV B6tiv ivtbg trjg 6v^7Ctd)6B(X)g tcbv bi^BCOVj 
dvB6tQa^^Bva (paCvBtai xad^dTCBQ iv totg i7CiTci8oig Tcal 
xvQtotg ivoTCtQOig^ o6a Si i6tiv ixtbg tfig 6v^jct(D6B(Dg^ 

20 xad^diCBQ i6tiVj xal (paCvBtai. 

i6t(o xoiXov ivoTCTQov tb AF^ Sftftcc 8h tb -B, btlfBig 
8h dvaxlco^Bvai al BA^ BF^ 6v^7Ct(o6ig 81 aitatv inl 
tb Z, vil;rj Ss t6 tB AE xal tb KNj xal tb (ihv KN 
ivtbg tfjg tov Z 6v(i7tT(o6Bcog ^ tb 81 AE ixtbg tf^g 

25 6v(i7Ctd)6BG)g. ovxovv ix^krid^Bi^&v t&v 6il;Ba)v xad^dbcBQ 
iv totg iniTciSoig xal xvQtotg iv6ntQ0ig (paCvstac tb 



2. 8\ To] 8\ i7tl7t8$ov x6 m. icvocyLXuG^^riG&v v. 7. 

T<J(alt.)] xh 8i m. 9. t'] ts' Vv. 12. Itfrco] ^axm nXdyvov m. 



CATOPTRICA. 



303 



sit oculus B, longitudo autem obliqua ^Ey spe- 
culum autem jiF. itaque radiis refractis z:/ in ^, 
JB in r adparet, et sicut re uera se habet, etiam uide- 
tur esse, propius propius, longinquius autem longin- 
quius. 

10. 

Longitudines obliquae in speculis conuexis, sicut 

re uera sunt, ita adparent. 

longitudo sit E/d, oculus 
autem B, speculum autem con- 
uexum AF, et radii ad E, /1 
refracti. reliqua uero eadem 
sunt. 




11. 

Altitudines et profunditates in speculis concauis, 
quae intra concursum radiorum sunt, sursum deorsum 
uersae adparent, sicut in speculis planis conuexisque, 
quae autem extra concursum sunt, sicut sunt, ita 
etiam adparent. 

speculum concauum sit ATy oculus autem B, radii 
autem refracti BA^ ^T et concursus eorum in Z, alti- 
tudines autem AE et KNy KN intra concursum in Z, 
^ E autem extra concursum. itaque radiis productis, 



14. tcc di — ccbtdli xal rj &7f6dsi,^i,s (pavSQci' d^oioc ydQ 
iati ty iv tolg iitv^tsSoig ivoTttQOig m. 15. icc'] i^' Vv. 

17. ftaV] fti} M. ivt6g^ i%t6g M. av^Litt&GBGig^ ntmasoDg^ 
supra scr. avy,, m. 20. ^attv] %6tv M. 22. avy,7tt€oaig\ 

tfviint<hasLg Vv. 23. t6 (pr.)] tov m. 24. tov] om. m. 

Z] ins. m. 1 V. 25. t&v 6il)Soiiv] om. Mm. 26. iv- 

6ntQ0ig] iv^ittQOig t&v 6il)Sfov m, iv^-xt^OKi i^niv^^^^. 



304 



CATOPTRICA. 



^€v K hcl tov M, ro S^ N ht\ xov A* &6tB &v£6tQan- 
(idva (paivstaL, ndXiv i%l tov ixtbg tfjg 6v(i7Ctc}6€a)s 
vil^ovg (paLVstai tb (ihv /1 inX tov Hy ro 8^ E iTtl 
tov ®, G)g i%£L^ ovtfog (paCvEtai. 
6 Tcdkiv pdd^og fihv tb ^E xal KS^ ivonxQOv 8\ 
Ttolkov tb AF^ {i(i(ia 8} tb B^ StlfSLg 8h avaxX(0(i£vai 
xal 6v(i7ti7Ctov6aL xaxd xb Z. oixovv ixpXrjd^SL^cbv 




x&v otl>£(ov 6(ioi(og xa (lev K^ (paCvBxaL dv£6xQa[i(i£vaj 

xb (ilv K xaxd xb F, xb 8h & xaxd xb A^ xad^dTCEQ 

10 iv xotg iTCLTceSoLg xal xvQxotg ivdnxQOLg^ xd 8% /d^ E^ 

xad-diCBQ xal £6xlv^ xb (isv E xdxco xaxd xb A^ xb 8% A 



av(o xaxd xb F. 



fc/3'. 



Td TckdyLa (iTJxrj aTcb xcbv xoCkcov ivoTCXQCsv^ Z6a 
15 ii^v ivxbg xrig 6v(i7CX(o6£C3g x£txaL x&v o^acov, xad"- 

1. Toi) (utrumque)] ro M. ScvtsaTQcc^nLivcc M. 3. tov 

t6 M. 4. TOV] TO M. COff] «(TTf COff m, COS OVV M. OVTGii 

ovTOD m, ovrco yicci M. 5. trj' Vv. Ttdhv — 12. T] xa 

^ttI tcop ^ad^div oftoicog T) a-uxTi ^cxiv 6.'it^S%\,&,\,<5 m. 6. AF] 

A^ M. 9. r] A M. 11. i^Gxi^. A\U^. YV vs:\ 
i^' Yy. 15. xsrtat] ^EcoQEixai U. x»v Xi'xi?%ui'» •*.^\%iiv ^. 



CATOPTRICA. 



305 



sicut in specuKs planis conuexisque, iiC in M adparet, 
N autem in A. quare sursum deorsum uersae ad- 




parent. rursus in altitudine extra concursum posita ^ 
in H adparet, E autem in @; quare, sicut est, ita 
adparet. 

rursus profonditas sit ^E et K®, speculum autem 
concauum AF, oculus autem Bj et radii refracti et 
in Z concurrentes. itaque radiis productis similiter 
puncta Ky @ sursum deorsum uersa adparent, K in 
r, & autem in ^, sicut in speculis planis conuexisque, 
^, E uero, sicut sunt, E inferius m Jt, ^ autem 
superius in F, 

12. 

Longitudines obliquae in aTpeQ,\3Si^ ^^^jiSiasQ^^ ^^j^iaR^ 
nitra concursnm radioruixi poa\V.ae «vmA», «v^^j^ ^nss:^^ "^ 

EuclideB^ edd. Heiberg et Menge. TLX. ^ 



306 CATOPTRICA. 

dneQ ItftLv^ ovT(o xal ipalvBtaL^ o6a d' ixt6g^ Avts- 
6tQaiiiidva. 

i6t(o yd^Q iv/fitri [ihv nX&yia t& EJ^ SK^ xotXov Sh 
ivoTCtQov to AF^ ii^(ia Sh ro 5, SrffSLg Sl dvaxXAfievai 
6 xal 6v^7C^7Ctov6aL xatic to H at BAjd^ BFE^ xal tb 
^hv &K TckAyLOV firlxog l6t(D ivtbg trjg 6v[iXTdt6scog 
tfjg H^ tb Sh AE ixt6g. oixovv td fihv 0, K xatd 
(fi^LV (paivetaL^ xad^dsceQ iv totg ixLTciSoLg xal xvQtotg 
iv6jctQ0Lg^ t& Sh E^ A dvte6tQayi,ydva' tb ^hv y&Q A 
10 i%\ tov A (palvetaL^ ib Se E ixl tov F. 

l/. 

^vvat6v i6tL SLa jcXbl^vgjv iv67CtQ(ov imjciScov 
ISetv tb aift^. 

i6t(x)^ 8 Set d^pd^rjvaL^ tb Ay (i^^a Sh tb -B, ivojctQa 

16 d£ tQLa td rj., ^E^EZ. ^x^(X) Sii xdd^etog dicb roiJ B 
iTcl tb rj ivoTCtQOv fi BF^ t6ri Sh ii Br ty FU^ xal 
TcdXiv djcb tov A iicl tb EZ xdd^etog fi AZ^ xal rg 
AZ t6ri 7] Z&^ xal dicb tov & iicl ro AE ivoTCtQov 
xdd^etog %'9'(o fi &K^ xal icta tf] &K t6rj ii KA^ 

20 xal dnb tov A iicl tb 2J ijce^evxd-(o ij AMtSU^ aTcb 
Sh tov M iicl tb & fj MP&^ i%e%evx^(o6av S\ xal 
aC AP^ BS. inel ovv t6r] i6tlv ^i BT t^ rU^ xal 
dQd^al at TCQbg ta F yavCaL^ Svo Sii at BF^ FO Sv6l 



5. H] N V. BAJ] AB, AJ M. 7. ra] r6 m. 9. ra] 

(palvstaL yccQ tb iikv S v,octa tb A, tb dh K ticctcc tb F, t6 m. 

Scvt£6tQcc(nievov m. 11. ty'] %' Vv. 12. iativ v. 16. 

tcTi — tfj] yioX tfj BF tari ^*^^® rj m. 17. Sc7f6] ini v. troti] 

corr. ex t6 v. A] postea ins. m. t6] tijv M. EZ] ZE 

hoTtTQov m. ^a-^aTOs Tix^co i^a.. Ift. fctv fe<a m, ^E] 

in ras. m. 19. ^ct©^ om. m. t\ ^V5X\,::^\ Wci ^^^ •*\^« 

20. imSs^dx^d^^o M. ^MBS'^ AMS^^. ^V xS\ x^-v ^L.^ , 



CATOPTRICA. 



307 




etiam adparent, quae autem extra sunt, sursum deor- 

sum uersae. 

sint enim longitudines obliquae E^y &K, speculum 

autem coneauum ^T, oculus autem B, et radii re- 

fracti et in H con- 
^'^ currenieaBA J,BrE, 

et longitudo obliqua 
@K intra H con- 
cursum sit, jdE au- 
tem extra. itaque &jK 
secundum ueritatem 
adparent, ut in spe- 
culis planis con- 
uexisque, E, A uero 

sursum deorsum uersa; /i enim in A adparet, E 

autem in r. 

13. 

Fieri potest, ut idem compluribus speculis planis 
cematur. 

sit A id, quod cemi oportet, oculus autem sit B, 
et tria specula TA^ ^E, EZ. iam a B ad F^ spe- 
culum perpendicularis ducatur BF, sitque Br=rUy 
et rursus dih A2AEZ perpendicularis AZ^eiZ& = AZy 
ei ^ & 2lA AE speculum perpendicularis ducatur &K, 
et sit KA = &K, et ab ^ ad 27 ducatur AMS2J, 
ab M autem ad & recta MP&, et ducantur etiam 
AP, BtS, iam quoniam BF«= FU, et anguli ad F 
positi recti, duae BF, FO duabus 27 JT, FO aequales 



et V, sed corr. 
om. Mm. 



23. Tc5] x6 mY. r$\ T^ te.. ^-ocX-v"^^ 



«i.^^- 



308 CATOPTRICA. 

tatg Ur^ r0 t6ai al^Xv BTcatiQa ixatiQCC^ xal ycavia 
fl i)%o BFO dgd^ii ov6a ytavla tfj {mb UFO d^d^ 
ov6ri t6ri i6tCv^ xal at Xoncal yfoviai talg XoiJtatg 
ycovcacg t6aL i6ovtai^ ixp^ cig at t6ai TcXsvQal ifTCo- 

6 t€LV0v6iv^ ii fihv XQbg tm B ycovCa tri Jtgbg t& 27, f^ 
S\ S yovCa tfj T. alV i\ T tfi N i6tiv t6rj' xata 
xoQvtf^v yccQ' &6tB t6rj i6tl xal ij N yan/Ca xfi S' 
fl aQa BtS Z^ig avaxXa6%^ifj6Btai inl ro M, TCaXiv 
ijtsl t6ri i6tlv fj 0K tfj KA^ xal dQd^al Sl at TCQbg 

10 rp jRT, t6i] i6tlv fi O yovCa ty 11. avaTtXatai &Qa fj 
avtii Hilfcg ij BSM inl tb P. Sict ta aircic Sij Ttal 
inl tb A Si& tb t6rjv Blvai tijv i}7cb ZPA ycovCav tfi 
i)7tb EPM biioCog tatg Xoijtdtg d7toSBC^B6LV. 6q& &Qa 
ij &7tb tov B o^^atog oipig ro A Sid t&v tQL&v iv- 

15 oTttQcav ovtov iTtLTtiSov t&v r^, AE^ EZ. 

lS\ 

"E6tL Sb xaCj Sl^ o6iov ccv tLg iTtLtd^rj ivdittQov 
ijtLTtiScov^ lSblv tb avtd' Sbl Sb Tcatd tbv dQLd^^ibv t&v 

ivOZtQOV TtoXliyOVOV l667tXBVQ6v tB Xal i60yG}VL0V 

20 ^vvL^ta^d^aL Sv6l TtXBCovg i^ov TtXBVQag t&v iv67ttQa)v. 

i6tG) yaQ^ ^lv dg^d^flvaL Sbl^ ro A^ o^^a Sb tb B^ 

xal iitB^Bvx^G) i] AB^ xal ditb tfjg AB dvayByQdtpd^a) 

noXvycovov l667tXBVQ6v tB xal l6oyG)VLOv Svo TtXBVQ&g 



1. 2r, r^] rs, 2^ m. r^] re m. 2. bt^ 

B rS m. oQ&ri et Sq&jj] ante ^ ras. 1 litt. V. 2r0 

2rS m. 3. iari Mm. ycoviai] yoaviuig M. 4. 'bno- 

xivovGiv V. 5. xm (pr.)] corr. ex x6 m, x6 v. x& (alt.)] r6 v. 

6. ^] $ m. ' T (alt.) — i67\\ T ycovia rj N tari ^«^^^ m. 

7. iaxiv Vv. tSl] ^ m. 9. Ss] om. m. 10. rc5] x6 v. 
K] K ycaviai m. 11. B^M] B^ M. 14. JB] e corr. m. 
XQL&v] y M. 16. 18'] xa' Vv. 17. ^axiv V. imtd^fj] 



CATOPTRICA. 



309 




sunt singulae singulis, et L BF^ rectus angulo 2JrO 
recto aequalis est, et reKqui anguli reliquis angulis 
aequales erunt, sub quibus latera aequalia subtendunt, 

uerum L T = N] 
nam ad uerticem 
positi sunt; quare 
etiam L N = tSl. 
itaque radius BS 
ad M refringetur. 
rursus quoniam 
®K=KA, 
et anguli ad K po- 
siti recti sunt, erit 

L O = n. itaque idem radius BSM ad P refringitur. 

eadem de causa etiam ad j4, quia, ut in reliquis demon- 

strationibus, demonstrari potest, esse L ZPA = EPM. 

ergo radius oculi B tribus speculis planis n^, jdE, EZ 

cernit A. 

14 

Licet autem etiam, quotcunque specuHs planis 
iubemur, idem cemere; oportet autem secundum nume- 
rum speculorum polygonum aequilaterum et aequi- 
angulum construere latera habens duobus plura speculis. 

sit enim A id, quod cerni oportet, oculus autem By 
et ducatur ABy et in AB polygonum aequilaterum 
et aequiangulum construatur latera habens duobus 



iicixdh,BiBv Mm. 18. uM' M'] Ss M. di] drj Mm. 

&Qid'ii&v mv. 19. Tf] supra scr. m. 20. ^xav v. 22. 
xal &vccyByqdq>^ei &nh r^s AB m. 23. xe\ «vsj^Tik* ^6Rx. ^, 



^ 



310 CATOPTRICA. 

TckBLOvg l%(yv t&v ivixxQfov xccl i6r(D rb AB/i Tcokv- 
ythvLOv^ xal slXtlipd^iD tb xivtQOv tov xijkXov tov y^a- 
fpofiivov xsqI tb noX^iyfovov ro ©, xal hc^ aiytov 
his^six^^^av al ©r, ®E^ ©^, @B, ®A iitX tdcg yco- 
5 viag^ Tcal HQO^xsC^d^CD^av ivoxrga hccTCsda XQbg d^d^&g 
talg ins^svyiLivaig, insl ohv t6ri i6tlv ii ZA ycnvCa 
ty NK' dQdij ydg i6xLV ixatiQa' &v fi N ty A t6ri 
i6tCv^ XoLxij aQa ii Z t^ K t6ri i6tCv, &6ts i^ dvd- 
xXa6ig ty]g BF Zil;s<og iicl tb A i6tai* 8ih ydq t6a)v 
10 yavL&v at dvaxXd6SLg yCvovtaL, 6fioC(Dg Sh SsL%%^6ovtai 
xal at XQbg totg A^ E 6rjfisCoLg ycavCaL t6aL at ngbg 
tolg ivdictQOLg, '^ &Qa dicb tov B ofifiatog oilfLg dva- 
xX(oiLivri xal jCQ067ts6ov6a XQbg icdvta td ivoTCtQa ^^sl 
iicl tb A, 

15 LS\ 

"E6tL dh xal dLd xvQtcbv ivdTCtQcjv xal Slcc xoCkov 
Idstv tb aitd, 

i6to ydQ^ dsL iSslv^ ro A, o^^a Ss ro B, xal 

b^oCcjg dvaysyQdcpd^Gi icokvycnvov 1667cXsvq6v ts xal 

20 l6oy(ovLov tb ABFAE^ xal jCQog totg T, Aj E 6YjfisCoLg 

i6tG) SvoTCtQa iTcCTCsSa^ dcp^ av bQcctaL ro Aj xad^djcsQ 

SiSsLXtaL^ xal XQO^xsC^d^ca tovtOLg xotXa -^ xvQtd iv- 



1. ^%o)v V, sed corr. t&v] xcbv iTCitax^ivtoav m. %ai 
— TtoXvy^vtov] tb ABFJE m. 2. yQocqjonivov] om. m. 3. 
TtSQll iiti Mv. 7toXvy(ovov — avrov] ABFJK TtoXvyoDvov 

TtSQiyQacpoiiivov xal ^ffro) tb xal &7tb rot) @ nivtQOv itQbg 
tccg tov ABFJE ■jtoXvymvov ycoviocg m. ytoXvyoovov] TtoXv- 

y&viov M, et V, sed corr. 4. al] e{)^6toa ocl M. @A, ©JB, 
Or, @J, @E m. iitl tocg yaiviocg'] om. m. 6. iTtiSsvyfiivaLg 
Vv; @r, @J, @E m. 7. NK] K M, KN m. 9. J ^trrat] 

d-n M. 12. dn^atog] V, om. Mmv. 13. 7tQoa7ts6ovca] TtQoa- 
nLjttovoa m. 15. is'] x^' Vv. 16. iv6ntQ(ov — noiXiQv] 



CATOPTRICA. 



311 



plura speculis, sitque AB^ polygonum, sumatur 
autem ® centrum circuli circum polygonum descripti, 
et ab eo ducantur ©T, ®E, ®^, @B, @A ad angulos, 

speculaque plana ad rectas 

ductas perpendicularia ad- 

ponantur. iam quoniam est 

L Z + A =N + K (nam 

uterque rectus est), quorum 

L N== Ay erit etiam LZ — K. 

quare refractio radii B T ad ^ 

fiet; sub aequalibus enim 

angulis refractiones fiunt. et 

similiter demonstrabimus, etiam angulos ad A, E puncta 

positos ad specula aequales esse. ergo radius oculi B 

refractus et ad omnia specula adcidens ad A ueniet. 




sM^. 



15. 

Licet autem etiam speculis conuexis concauisue 

idem cemere. 

sit enim A id, quod 

cemi oportet, oculus autem 

By et eodem modo poly- 

gonum aequilaterum et 

aequiangulum construatur 

ABFAEj ad puncta autem 

r, A^ E specula sint plana, 

imde cemitur A, sicut de- 

monstratum est [prop. 14], 




rj noiXmv iv6%tQ(Qv m. 19. &vayByqa,(pQ'(o dtboiag m. 20. 

ABFJE] corr. ex ABJE m. 1 Y. 21. &(p^] itp' M, $i' m. 
22. %ai] om. M. 



312 CATOPTRICA. 

ojcxQa xat& tccg iap&g t&v otl;e(ov. oixoijv t6fj i6tlv 
il il\v Z tri ®^ ii 8% K tfi A' Uri &Qa ij KZ C^fj i6tl 
tri ®A. &vaxXa6%'Tfj6atai, aQa ij iitlfig &%() tov ;cvprov 
ivdjttQOv tov r iTcl ro jd xal aTch tov A iid tb E 
6 xal aTcb rov E inl ro A, fpavsQbv ohv^ oti xal xvq- 
t&v tj xolXcov ovtcDV icTcdvtmv xal avafi6fi,Lyfi,iv(ov 86tLV 
ISelv ro a^rd. 

L^\ 

^Ev tolg iTCLTciSoLg ivdTCtQOLg exa6tov t&v 6Q(Ofiiv(ov 
10 xatic tijv ajcb roi; bQcofiivov xdd^stov &QataL. 

i6t(o ivoTCtQov ijCLTceSov tb TA^ ofifjLa Sh ro B^ 

^ ' bQcjfisvov Sh ro A^ xal i6tco xdd^stog ij aTcb tov bQG)- 

liivov iicl tb ivoTCtQOv fj AF, oifxovv iTcel imixsLto 

iv totg (paLVOfiivoLg^ ZtL xataXrj^pd^ivtog tov t6jcov 

15 rov r ov% bQataL tb A^ tb A ccQa 6(pd"ifj6BtaL irc^ 

s^bd^sCag tfj AF. aXka Sij xal in^ sv^siag trj B^ Ztf^SL^ 

i xatd tb E ccQa' iTcixsLtp yuQ ij^tv tb sv^v^ ov tb 

^i6ov totg ccxQOLg iTCLTCQO^d^st' &6ts svd^sta i6taL ij AE 

;cal fj BE, 

20 fcg'. 

^Ev totg xvQtotg iv67CtQOLg sxa6tov tSiv 6QC3iiiv(ov 
xatd f^v dicb rov bQO^ivov slg tb xivtQov tr^g 6(paL- 
Qag dyofi,ivriv svd^stav bQcctaL, 

s6ta) xvQtbv ivoTCtQOv ro FA^ S^^a Sh ro B, (iifLg 



2. 7) (pr.)] eras. v. Heri — ^- ®-^] ^^ tjj A@ hri icriv m. 

2. ictiv Vv. 5. xa£(alt.)] oin. Mvm. 6. %ai'\ ij m. &va- 
(is^iy^ivov m, sed corr.; &vaii£ii7iYii,ivoi}v v, sed corr. 8. tg'] 
xy' Vv. 10. rov'] r&v M. 13. vTCs^Sito] 'bitoY.sitai m. 14. 
(faivo[Lsvoig] oQoig m. 16. AF] JF Mm. BJ] BA Mm. 

17. natd] iisrd M. ^^a] om. m. vTCSKSito] 'bit^v.svt ai m. 



CATOPTRICA. 313 

iisque adponantur in punctis contactus radiorum spe- 
cula concaua conuexaue. itaque /. Z = ©, K = A. 
itaque /.-K+Z=0-t"^- quare radius ab speculo 
conuexo f ad ^ refringetur, a ^ autem ad E, ab E 
autem ad j4. ergo manifestum est, etiam speculis 
conuexis concauisue omnibus et mixtis fieri posse ut 
idem cematur. 

16. 

In speculis planis omnia, quae cernuntur, secundum 
rectam ab eo, quod cemitur, perpendicularem cemuntur. 

sit r^ speculum planum, ocu- 
lus autem B, cematur autem A^ 
et ab eo, quod cemitur, ad spe- 
culum perpendicularis sit AF. ita- 
que quoniam in phaenomenis sup- 
positum est, loco F occupato non 
cerni ^, A in recta AF producta 
cemetur. uerum etiam in radio 
B^ producto cemitur. ergo in E cernitur; supposui- 
mus enim, rectum esse, cuius partes mediae extremis 
officerent; quare AE^ BE rectae erunt. 

17. 

In speculis conuexis omnia, quae cemuntur, secun- 
dum rectam ab eo, quod cemitur, ad centrum sphaerae 
ductam cemuntur. 

sit FA speculum conuexum, oculus autem B, radius 



tb s^b^v] siO^v m. o5] bIvcii o^ m. 18. %6xai\ iativ M. 
AE] BE m. 19. BE] JE m. 20. tj'] xd' Vv. 24. 
&ij)Sig V, sed corr. 




314 CATOPTRICA. 

dl 17 Bjd &vaHX<ofiivfj inl ro ^, Tcal bQ&^d^cn th A^ 
xsvtQOv dl tfjg 6q>aiQag &ro r6 Z, xal ^«g^iJ^O-cD ^ 
-^^Z, xa^ ixPepXi^^d-oa '^ B^ "6il;tg hX th E. oi>7covv 
iTtsl 'bnixBLto iv tolg q>aLVOfiivoLg^ Ztt xatalr^ipd^ivtog 
6 rov r th A oix hgataL^ dq>d"il6£taL &qa i%^ sid^SLag 
„ tfi AF xat& tijv 6vfiPa6Lv rijg B^ &tlf6<og ocal {&7ch] 
tfjg AF iTcl tov Ej xad^djcsQ hcl totg iitLTciSoLg. 

Lrj\ 
^Ev totg xotXoLg iv6yctQ0Lg sxa6tov t&v hQCjfiivov 
10 xata tijv oTch tov hQOiyLivov slg th xivtQOv tflg 6q>aCQag 
iyo^ivrjv sid^stav hQ&taL. 

i6t(D xotXov ivoTCtQov th Fjd^ (iil^Lg dl ivaxkcoiiivi] 
rj BF ijcl th A 6q6[isvov^ tfjg dh 6q>alQag xivtQOv 
i6t(o ro E^ xal aTch tov A iTcl th E iTCs^s^dx^fo svd^sta 
15 xal ixfispiTJ^d-c:). oixovv iTCsl iicixeLto iv totg q^aLvo- 
laivoLg^ StL xaxaXri^pd^ivtog tov t67Cov tov A th A 0%)% 
hQ&taL^ &0ts (paCvataL iic^ svd^sCag tfi AE^ 6(p%"if^6staL 
, aQa xata tijv ev^fiokijv tfjg A^ sv%^sCag xal tfjg BF 
otl^scog xar(i ro Z. 

20 td-\ 

^Ev totg iTCLjciSoLg iv^ictQOtg t& Ss^l^: &QL6tSQic 
(paCvstaL xal ra aQL6tSQ& ds^L& xal th sCSaXov t6ov 
tp dQCj^ivc)^ xal th a7c66tYj^a th ajch tov iv67CtQOV 
l6ov i6tCv. 



1. A (alt.)] AE jjiy JE 'M.. 4. vnsyiSLro] i37i6%Htai m. 

tfaivoiLSvoig] OQOLg m. 5. ov^ dQ&tai tb A m. 6. rj] 

tfig Vv. Tijv] om. M. 6^116'] om. m. 7. ini (alt.)] iv Mm. 

8. 171'] x«' Vv. 12. rz^] AFM, rj 6Va Sl tb B m. 14. 

sifd^stoc] s^dS^sia ij AE m. 15. •uit^v.tixat iv xot^ oqoi^ m. 17. 

^oilvsad^ccL M et e corr. m. "i Y. xti\ tf\§ "^^^m. \^. •*M.xa\ 



CATOPTRICA. 



315 




autem B^ ad A refractus, et cematur A, centrum 
autem sphaerae sit Z, et ducatur AZ^ producaturque 

radius B^ ad E. itaque 
quoniam in phaenome- 
nis suppositum est, F 
loco occupato non cemi 
Ay in recta A F producta 
cemetur, ubi BA^ AF 
concurrunt, scilicet in 
E, sicut in planis. 

18. 
In speculis concauis omnia, quae cemuntur, secun- 
dum rectam ab eo, quod cemitur, ad centmm sphaerae 

ductam cemuntur. 

sit r^ speculum concauum, 
radius autem BF ad A, quod 
cemitur , refractus , sphaerae 
autem centrum sit E, et ab A 
ad E recta ducatur et pro- 
ducatur. itaque quoniam in phae- 
nomenis suppositum est, loco 
^ occupato non cemi A, ita ut necessario m AE 
producta adpareat, in puncto concursus rectae A^d 
radiique BF cemetur, h. e. in Z. 

19. 
In speculis planis partes dextrae sinistrae adparent, 
sinistrae autem dextrae, imagoque ei, quod cemitur, 
aequalis; et distantia a speculo aequalis est. 

supra scr. m. r6 (alt.)] l knk%^t xb ElScaVo-v m. ^«^- V^Ov.-^ 




316 CATOFrRICA. 

€6tco ijtLTtsSov evoTCXQOv xo AF^ ofifia dh rb 5, 
oifBLg S\ at BA^ BF ivaxkco^svaL iTcl td: E^ A^ 6q6' 
fievov Sh i6t(o to EA^ xal aTcb tS)V E^ A ijcl tb ev- 
OTCtQOv xdd^etoL Hxd^co^av at EZ^A® ocal ixfiefikrl^d^ca- 
5 6av^ iKfiefikrl^d^co^av Sl xal at BF^ BA oifeLg xal 6vfi- 
7CLJCtetG)6av tatg xad^etOLg xatd: t& K^ A^ xal ijce^evx^(o 
il AK, oixovv (paCvetaL tb ^lv E ijcl tov K^ tb 8\ A 
iicl tov A' tovto y^Q XQoeSeCxd^Yj. t& aQa &QL6teQ& 
SeiL& (paCvetaL xal t& SeiL& &QL6teQa. xal iicel t6ri 

10 i6tlv ij imb t&v KFZ ycovCa tfl imb t&v ZFEj xaC 
el6LV ^Qd^al at TCQbg tp Z, [67j av eHrj xal ii ZK tri 
ZE. Slo: t& aitcc xal i^ A® tfi ®A. t6ov &Qa tb 
ajc66tri^a^ 8 aicexeL &jcb roi) iv67CtQOv tb EA^ rcl, o 
ccTcexBL ro etSoXov ro KA, xal t6ov tb bQthfievov tb 

15 EA t& eiSdikco rp KA Slo, tb t6riv elvaL ti^v [ihv EZ 
t^ ZKj tijv Sh A® tfl ®A^ xoLvijv Sl xal TCQbg dQd^ag 
f^v ®Z. 

X . 

jBi/ ror^ xvQtotg iv67CtQOLg ta aQL6teQ& Sei,La cpaC- 
20 vetaL xal tSc Se^La aQL6teQ&^ xal ro a7c66tri^a aTcb tov 
iv67CtQOv tb etScoXov eka66ov ix^L, 

e6tc3 evojctQov xvQtbv ro AFj xevtQov S^ tf^g 
6<paCQag ro 0, S^fia Sl ro 5, S^ctg Se at BA^ BF 

2. dQmiisvov — 3. EJ] om. m. 4. yid&stog V, corr. 

m. 2. 6. Br, BA] EZ, BF, BA M. 6. tatg] roZs M, taig 
JA, EK m. 10. T&v KFZ] FKZ m. x&v (alt.)] om. m. 

ZFE] corr. ex ISIFE v; BAE, supra scr. FZ, M. 11. x& 
to V. Lari] Ccri ^Q^ Mvm. civ strf] J^axoci m. 12. %ai 

di} nai m. 0A] corr. ex ©^ m. 2 V, corr. ex JA M. 13. 
(alt.)] cS vm et supra ecr. m. rec. V. 14. t6 (quart.)] rco M, 
et V, sed corr. 15. stdmXco] 6qo)^svco M. 16. dG] 0d m. 

18. %'] ^r Vv. 20. &7t6 — 21. Ix^C] ocTtsxSL rb stSaXov 

ScTtb rov iv^TttQOv, ^Xa666v ian rov cc7toarr]^arog , ov &nsxsi rb 



CATOPTRICA. 



317 



sit AF speculum planum, oculus autem B, radii 
autem BAj BF Sid Ej ^ refracti, cernatur autem E^^ 
et ab Ey ^ ad speculum perpendiculares ducantur 
EZy ^S et producantur, producantur autem etiam 

A_ K 




radii B Ty BA et perpendicularibus concurrant in K^ A, 
et ducatur AK E igitur in K, jd autem in A ad- 
paret-, hoc enim antea demonstratum est [prop. 16]. 
ergo partes sinistrae dextrae adparent, dextrae autem 
sinistrae. et quoniam LKrZ = ZrE, et anguK ad Z 
positi recti simt, erit etiam ZK = ZE. eadem de 
causa etiam ^&= ®A. ergo distantia, qua EA a 
speculo abest, aequalis est distantiae, qua imago KA 
abest. et quod cemitur EA, aequale est imagini KA, 
quia EZ = ZKy A®=®A, et ®Z communis et 
perpendicularis. 

20. 

In speculis conuexis partes sinistrae dextrae ad- 
parent, dextrae autem sinistrae, et imago minorem 
habet distantiam a speculo. 

sit AF speculum conuexum, centrum autem 
sphaerae &, oculus autem 5, et radii BA, BF B,d A, E 



'OQdoiisvov, Ttal rb Bt8(oXov ^Xaaadv ^m rov dgca^svov m. 23. 
B] B, bQ&iLSvov ds rb JE, m. BA, Er\ ^r^Y^k^. 



318 CATOPTRICA. 

&vaxX(afisvat iitl xa ^, E^ hQcoiisvov Sh tb ^E^ Tcal 
&'jih tov xBvtQov V{%%^6av ijiX tct ^, E at 0^/, SE^ 
xal ix^B^Xif^^Q^fo^av at Sifscg iTcl t& Z, ff, xal ijts^siix^f^ 
tb ZH stSoXov, oitxovv tb fihv /1 tpaivstai iTcl tov H^ 
5 tb Sh E ijtl tov Z. tit aQa Ss^iSt &qi6tSQk (paCvstai 
;cal tic &QL6tSQ& Ssi,L&, kiy(o^ 5tt fiSL^cov i6tlv 'fj EA 
trjg AZ, iix^^ V^Q **^ ''^^ ^ ifpajttofiitnj tijg ^sql- 
(psQsCag ii PAK, ijtsl ohv at BA^ AE JtQbg tijv TtSQL- 
(piQSLav t6ag 7tOLOv6L ycavCag Sl& tijv &v&xXa6LV^ iq)- 
10 &7ttstaL Sl i^ KAP^ SCxa &v stri tstfirjfiivri i^ ijtb t&v 
EAZ ycovCa, xal &iipXst& i6tLv i^ K yovCa' (isC^cov 
aqa fi EK trjg KZ' TtoXXdi fiaUov ij EA trjg AZ. 
iXa66ov aQa &%ixsL tb stSc^lov tb ZH &7tb tov iv- 
dTttQOv^ fist^ov Sh tb bQcofisvov tb EA, 

16 xa\ 

'Ev totg xvQtotg ivoTCtQOcg tb stSaykov sXa666v i6tL 
t&v 6QG)^iva)v, 

i6tc3 y&Q xvQtbv svoTCtQOv tb AOr^ &^^a Ss tb Bj 
'd^stg Sl &vaxX6^BvaL at BA^ BF iitl ra A^ E, ovx- 



1. ra] m, r6 VMv. dQm^svov — dE] om. m. 2. ©J] 
e e corr. Mm. 3. al] al BT, BA m. Z, H] H, Z m. 4. 
rov] xo M. 5. Tov] t6 V? 6. oxi] 8i] 8rt m. pusitov v. 

7. A] corr. ex H m. i(pa7txoiisvov M. TtSQKpSQsiag] 

GtpalQag m. 8. ^iE] E V. xriv itSQnpsQSiav] rg TtSQicpSQsla m. 

9. TtoLOvatv Vv. ycovlag noiovGi m. 10. xsxay^svi] v. 

x&v] om. m. 11. EAZ] AEZ M. xa£] M xfjg KA 

svd^sLag m. iaxiv — yoDvla] $1 ri vitb EKA, o^sta $s t) 'bitb 
AKZ m. pLStSov V. 12. iiaXXov] aQa iisiiav m. EA] 

corr. ex EJ V. 13. ^Xaxxov M. ZH] ZN v. U. jistiov 
— EJ] i]7tSQ xb EJ dQmpLSvov m. Post EJ add. mg i^fjg 

xovxo 8si%vvxai Mv. 15. v.a'] nri' Vv. 16. iv — 17. 6q(o- 
HsvoDv] xal dnoicog Ssix^riasxaL, oxi yial xb JE bQ&iLSvov itst^dv 
icxi xov HZ sMXov m. 16. iaxtv Vv. 19. BA, BT] 

JSr, BA m. 



CATOPTRICA. 319 

refracti; cematur autem ^Ej et a © centro ad ^, E 
ducantur ©^, SEy et radii producantur ad Z, ff, et 
ducatur imago ZH, itaque jd 'm H, E autem in Z 
adparet. ergo partes dextrae sinistrae; sinistrae autem 




dextrae adparent. dico, esse EA > AZ, ducatur enim 
per A arcum contingens PAK. iam quoniam BAj AE 
ad ambitum aequales angulos efficiunt propter re- 
fractionem, KAP autem contingit, i EAZ in duas 
partes aequales diuisus erit. et /. if obtusus est. quare 
EK> KZ, itaque multo magis EA^ AZ. ergo 
imago ZH minus a speculo distat; sed quod cemitur 
EAj maius est. 

21. 

In speculis conuexis imago minor est eo, quod 
cemitur. 

sit enim-^OF speculum conuexum, oculus autem B, 
• et radii BA, BF bA J^ E refracti. \W^^ E*A -^ 



320 CATOPTRICA. 

ovv ujcb xov KVQXOv ivdTCXQOv d^SiOQetxaL xh Ejd iv 
ycDVLa xri i)jcb ABF, xccQaTcsi^d^io Sii ivoicxQOv iTci- 
tcbSov xb AF aTCxdfJLSvov x&v Sil^scov xaxit xd: A^ F. 
ovxovv i5 8^^? ^ iLiXkov6a ISelv xb E aicb xov im- 
5 TciSov ivdTCXQOv o\)7t i6XLv ii BAE' oi) yk^ %olbl ym- 
vLag i!6ag TCQbg xp imTciSci iv67CXQ(p. o\)S\ fiijv xAa- 
^^•ri^BxaL inexai^i) xcbv A^ F. xexld^d^Gj yag^ sl Svvax&v^ 
xal e6xo rj BZE '6tl)Lg. t6rj ilcQa ij H ycovLa xy B 
Slcc xiiv avdxka^Lv. ii Sl @ fiSL^ov xrig NI^ ij Sh M 

10 xfjg H' &6xe xal ri M xf^g NI fieL^cov i6XLV' oxeQ 
dSvvaxov. ai)xii ydcQ ij I fieL^C3V xfjg M i6XLV' t67j 
ydQ i6XLV okri xri TCQbg xfj jceQLtpeQBLcc. ixxbg iScQa dva- 
xla^d^TJ^exaL xov A. xexld^d^co xal i6Tco r] BKE. 
b^oLcog Sh xal ij BAA 7ce6elxaL ixx6g. xb aQa EA 

15 vTcb [leL^ovog ymvCag d^ecjQetxaL dicb xov isCLTciSov iv- 
67CXQOV xfjg TCeQLexo^evrjg vjcb KBA i^jceQ dnb xov 
xvQtov. l'6ov Se iSeCid^ri (paLv6^evov iv xm iTCLTCeSo) 
ivoTCXQco. cpaveQbv ovv^ oxl djcb xov xvqxov iv6jcxQov 
xb etScokov eka66ov (paCvexaL xov bQO^ivov. 



1. ivoTttQov] iv67ttQov tov AOr m. 2. ABF] A ia 

ras. V. 3. td] t6 M. 4. i^ (alt.)] om. VMvm. iiilXovaaj 
lacun. M. E] mut. in. EJ m. 2 Y. 5. ^ativ i}] forat t} 

avtj] rg m. 7. v.sv.Xl6^(o Mm. 8. &q(x\ ccQa iativ m. H] 
vnb BZr m. @] vnb BZA yavia m. 9. 0] vnb BZF, 

postea add. yavia, m. ^st^ov v, fisi^av iati m. Nl] Vv, 
vnb BAZ m, N M. 7] Ss — 12. TtSQLcpsQsia} xal ij vnb BZA 
aQa (supra scr. m. 1) yavia pisi^av iatl tfjg vitb BAZ' SitSQ 
iatlv advvatov m. 10. H] N Mv. Nl] N V?M. pbst- 
^ov V. 11. /] N M. iLSt^ov V. r^ff] tov M. 12. iv,- 

rdff] ivtog M. 14. oftotooff] ft V. 8s] om. M. i%tbg 

Ttsasttai tov F m. 16. tfjg — KBA] om. m. 17. I^aov — 
18. ivoTttQG)] ^si^av yccQ 7] vnb KBA tfjg vnb ABT Y.aL m. 



CATOPTEICA. 321 

Bpeculo eonuexo sub augulo ABT spectatur. adpona- 
tur igitur speeulum planum AF radios tangens in A, V. 
itaque radius, qui E a speculo plauo cernat, uon est 
BAE; neque enim angulos aeqaales ad speculum pla- 
num efficit. neque uero radiua ille inter A, F re- 
fiingetur. refringatur enim, si fieri potest, et sii 




radius BZE. itaque propter refractionem erit l.H=&, 
est autem Z.®>JV+/, M > H. quare etiam 
Jtf>.W"+ J; quod fieri non potest; nam LI>M; 
est enim toti angulo ad ambitum posito aequalis. ergo 
radius ille eztra A refringetut. reiringatur et sit BKE. 
similiter autem etiam BAA extra cadet itaque E^ 
a speculo plano sub maiore angulo, scilicet L KBA, 
quam a speculo conuexo spectatur. demonstrauimus 
autem, id in speculo plano aequale adparere [prop. 19]. 
ergo manifestum est, in speculo conuexo imaginem 
minorem adparere eo, quod cemitur. 



18. favefi^v^ tfayiiv M. ojri'] om. i 



L9. Gmzov VL, 



322 CATOPTRICA. 

^Ev rotg xvQtotg ivdxTQOtg &7Co t&v ika666v(ov 
iv67CtQG)v ika66ova fpaCvBtai t&, stSooXa. 

i6t(a 6<patQa fisi^cov fihv ii AF^ iXa66G3v 8% i^ EA 
6 TCBql tb aiftb xivtQov tb ©, &^fia 8h tb 5, xal ixB- 
^Bvxd^cD 'fj BA0, xal djtb trjg 6q>aCQag avaxBxXd^d^c^ 
'6tlfig i5 BFA, Xiyc3j Stc 'fj dvaxla^d^ri^ofiivr] *6^ig dnb 
trjg iXd66ovog 6fpaCqag ixl ro A oiitB dta tov F tcb- 
6Bttac oiits ixtbg tov F. jCLTCtitco yicQ 7CQ6tBQ0v^ bI 

10 8vvat6v^ 8id tov F^ xal dvaxBxXd^d^c^ dxb tfjg ikd6- 
6ovog 6€paCQag iicl tb A xal i6tcD i^ BEA^ xal ijCB- 
^Biixd^co dTcb tov @ iTcl ro F xal ix^B^Xifj^d^Gi iTcl tb K. 
8C%a 8ii tB^Bt 'fj ®rK tijv vjcb tS)v BFA ycovCav Sid 
ro tijv BFA t6ag icoiBtv yovCag TCQbg tfj %BQi(pBQBCa 

15 8Ld f^v dvdxXa6tv. ^id td avtd 8% xal fj djcb tov @ 
iicl ro E ijCL^Bvyw^ivrj xal ixfikrjd^st^a 8C%a tBfiBt tijv 
ijcb BEA. tB^vitcj xal B6tco fj ®EZ. iiCBl (ibC^c3V 
i6tlv fj TCBQLBxofiivrj vjcb tcbv BFA trjg ijcb BEAj 
xal ij fj^C^BLa tfjg fj^L^BCag ^bC^cov i6tlv fj VTcb BFK 

*20 tfjg {)7cb BEZ. i6tL 8h xal ikd66(XiV' otcbq d^ivatov. 
ovx ccQa i^^sL 8Ld tov F rj dvaxXcj^ivri '6ipLg djcb 
tfjg ikd66ovog 6(paCQag. 



1. %P'] yid'' Vv. 4. fisliav v. 6. BA@] B e corr. m, 

J5 @A M. rflff] rfjg AF m. 8. rfjg] om. M. iXdrrovog M, 
XT m. 9. yao] supra scr. m. 10. iXdaaovog] sX m. 11. 
iTts^svx^o) — 12. r] imSsvx^staa (-cc e corr.) i) BT m. 12. 
licci] om. m. 13. t&v] om. Mm. 14. r6] supra scr. m. 

BFJ Hipiv m. 15. ds] 8fi M. 16. yiai] svd^slcc %cci m. 

i-apXTid^stacc] iyipaXXoiisiiri m. 17. insi] %al iiisi m. ^ist- 
lov V. 18. nsQisxoiLsvri] om. m. r&v] om. m. BFd 

yo)via m. 19. i} vnb BFK rfjg i^Liasiag rfjg vnb BEZ m. 

i} (alt.) — 20. BEZ] om. m. 20. Eariv Vv. iXdrroav M. 



CATOPTRICA. 



323 



22. 

In speculis conuexis a minoribus speculis minores 
adparent imagines. 

sit sphaera maior AF, minor autem EA circum 
idem centrum ® positae, oculus autem sit B, et du- 
catur BA@, et a sphaera refringatur radius BFJ. 

dico, radium, qui 
a minore sphaera 
ad ^ refringatur, 
neque per F ca- 
dere neque extra 
r. prius enim, 
si fieri potest, 
per r cadat et a 
sphaera minore 
ad ^ refnngatur 
et sit BE^, et 
a ad F du- 
catur recta et 
ad K producatur. QFK igitur angulum BF^ in 
duas partes aequales secabit, quia BF^ propter re- 
fractionem aequales angulos ad ambitum efficit. eadem 
autem de causa etiam recta a @ ad ^ ducta producta 
angulum BE^ in duas partes aequales secabit. secet 
et sit @EZ. quoniam L BF^ > BE^, erit etiam di- 
midius dimidio maior, h. e. /. B FK > BEZ. uerum etiam 
minor est; quod fieri non potest. ergo radius a mi- 
nore sphaera refractus per F non ueniet. 




21. 7} &tj)ig ccvocnXtoiiivt} m. rj] om. VMv. 22. iXdrtovo(i 
M, eX m. 



324 



CATOPTRICA. 



ijtoxsL^d^co dh %aXiv xa aiyx&^ xal ^ oacb xfjg iXd6~ 

60V0S 6q)aLQccg avaKkcofiBvrj S^t^ i^ BE^ ixxbg TtLTtxsxc) 

xov r, xal xafivBxcj 'fj BE xi^v (iSL^ova 6q)atQav Tcaxct 

xb Z. ii 8ii aTtb xov Z 
5 dvaxXcjfiivti oilfig ii BZK 

oi) 6v(i7tB6sixai x^ r^' 

xovxo ydQ dsdBLXxai. xfj 

aQa E^ 6vii7tL7tXBX(0 xaxd 

xb K. ij &Qa BZK otl^Lg 
10 dvaxla}(iBVYi djtb xov fiSL- 

^ovog svoTtXQOv 6Qa xb K^ 

xal rj aixri ^ BEK dva- 

xkcDfiBVti dstb xov hkd66o- 

vog ivdTtXQov 6Qa xb 
15 avxb K' xovxo 8\ iTtdvco 

B8sL%%"ri dSvvaxov. fieralv 

aQa 7ts6BLxaL xg)V F^ A 7] 

dvaKkco^Bvrj 8^tg dTtb xov 

ikd66ovog ivdTtXQOv i7tl xb 
20 ^. b^OLCjg ds 8sL%%"ri6sxaL 

xal ri dTtb xov sxsqov ^SQOvg xb avxb 7t0L0v6a. i7tb 

ikd66ovog aQa yovCag d^scoQstxaL xfjg 7tQbg x& B yvyvo- 

^Bvrjg d7tb xov ikd66ovog ivdjtXQOV ^7tSQ d7tb xov 

^SL^ovog. ska66ov aQa (paCvsxaL xb stSmkov dTtb xov 
25 ild66ovog iv67tXQOv. 

xy'. 
^Ev xolg xvQXolg ivoTtXQOLg xd stScola xvQxd q>aC- 
vsxaL. 




1. Si'] 8ri m. ildtrovog M, il m. 3. /iftjoi/a] AF m. 

4. Z (utrumque)] N m. 6. BZK] BZE M, BNISl m. 8. 

Ed] corr. ex E-4 m. 2 V, Ez/ 6v(j,7ts6slTca ri NlS! m. 9. K] 



CATOPTRICA. 325 

nirsus eadem supponantur, et radius a minore sphaera 
refractus BE^ extra F cadat, BE autem maiorem 
spkaeram in Z secet. itaque radius a Z refractus 
BZK rectae F^ non concurret; hoc enim demonstra- 
tum est [prop. 4]. rectae igitur E^ concurrat in K. 
radius igitur BZK a maiore speculo refractus K 
cemit, et idem radius BEK a minore speculo refractus 
idem K cemit; hoc autem fieri non posse supra [p. 322] 
demonstratum est. ergo radius a minore speculo ad ^ 
refractus inter F, j4 cadet. et similiter demonstrabi- 
mus^ etiam radium ab altera parte refractum idem 
facere. sub minore igitur angulo ad B eflfecto a minore 
speculo cemitur quam a maiore. ergo imago a speculo 
minore minor adparet.^) 

23. 

In speculis conuexis imagines conuexae ad- 
parent. 



1) In y praeterea est haec figura 
add. avtri iatlv i] 'bTtoyisnisvri '^arri . . . 




n m. BZK] BZE M, BNlS! m. 11. K] S m. 12. aif^ 
om. m. ij^&lt.y] om. Mm. BEK] BE^ m. 13. iXdttovog 
M, comp. m. 15. K\ IS} m. iicdv^o] &voyti^(o m. 19. iXdx- 

tovog M, comp. m. 20. ofioioig] fi V. $s} om. m. 22. 
iXdttovog M, comp. m. tcai] corr. ex t6 m, to v. iiivQ^.ifw^ 
Mm. 23. iXdttovog M. '25. iUttovos^. 'i^. •^'l\'^ ■^"« 



326 CATOPTRICA. 

i6t(o xvQtbv ivojttQOv tb AF^ Sfifta 8% tb jB, S^£^^ 
8a avaxlaiisvuL at EA^ EF iTtl tcc ^, JB, ^ dh ZE 
dvaxl(0fi6vri 8l[ iavtfis ktl tb E, ovkovv t&v titlfSCDV 
^ay t6tav ^iiv aldv tm fii^xai al jcoQQwtdtG) ^ ikd%L6tai, 
5 8a aC Tcatcc [li^ov^ &67taQ fj EZ. (paivatav aQa tov 
ivdntQov ayyvov iiallov tb E^ TtOQQC^tdtcD 8h rb B 
Tcal ro ^, &6ta olov xvQtbv q)aivatav, 

x8\ 

^Ev totg xockocg iv6%tQ0ig idv ijtl tov xavtQiw ro 
10 6(iua tad^^ aitb fidvov (paivatai ro Sfi/xa. 

i6tc3 xotXov ivojttQov ro AF^^ xivtQOV 8a aiftov 

ro 5, Hil^aLS 81 a[ BA^ BT^ B^. oixovv t6ri i^ E 

ycjvia tfj Z. 7]^aL ccQa dvaxXcofiivti ii BF iil^cg i%\ 

ro B, b^oicng 8\ xal ai XoiTtaL aiytb iiovov aQa 

15 oQccraL ro B, 



xa\ 



^Ev tolg xoCkoig ivoTttQoig idv iitl tfjg jtaQKpaQaCag 
d^fjg tb liiLiLa 'J) ^'lco ry]g TtaQicpaQaCag^ ov (paCvatai tb 



oftfia. 



20 i6tG) xoilov avoTttQov ro AFB^ xal ro oftftc^ xaC6d^c3 
aTtl tfjg TtaQKpaQaCag avrov ro 5, oil^aLg 8h JtQ067tLJtti- 
t(D6av aC BA^ BF xal dvaKaxkd^d^co^av. oifxovv fiaC- 
t,ov i6rlv ii ^av MS ycovCa rrlg K^ fj Sa EA trlg Z, 



2. EA] EA M. 3. §ocvtfjs] ocvtfjg M. 4. siatv Vv. ocl] 
corr. ex s v. TtOQQoardta) rov fisaov m. 5. xara] xam ro m. 
7} EZ] corr. ex ij E^ v, ivrocvd^oc [iBy leroci fiiv slciv ai dA, BFj 
iXocxiGrr} dh ij EZ m. 8. %$'] Xoc' Vv. 9. rov v,ivxQOv] 

t6 TiivrQov M. 12. tari] ^*^^ iorlv m. 14. Xoiitoci] BA xoci 
7i4/ ^fsig inl tb B '^'lovciv m. l^. y.e'\ !.§' Vv. 18. &ig] 



CATOPTRICA. 



327 



sit AF speculum conuexum, oculus autem Ey et 
radii EA, EF a,d ^^ B refracti, ZE autem secundum 

se ipsum ad E refringatur. ra- 
diorum igitur maximi longitu- 
dine sunt, qui maxime remoti 
sunt, minimi autem medii, ut 
EZ. quare E speculo propius 
adparet^ remotissima autem B 
et ^. ergo totum conuexum ad- 
paret. 





24. 

In speculis concauis si in centro oculus ponitur, 

ipse OGulus solus adparet. 

sit jir^d speculum concauuni^ 
centrum autem eius 5, et radii 
BJ, 5r, B^. itaque L E == Z. 
radius igitur BF refractus ad B 
ueniet [prop. 2]. et similiter 
etiam reliqui. ergo ipsum B solum cernitur. 

25. 

In speculis concauis si in ambitu uel extra ambi- 
tum oculus ponitur, oculus non adparet. 

sit AFB speculum concauum, et oculus B in am- 
bitu eius ponatur, radii autem adcidant BA, BF re- 
fringanturque. itaque LM + @> K, LE -\- A> Z. 
quare radii BA, BF 2A oculum B non refringentur; 



XB^^ M. xb oiL^a (alt.)] x6[ntM V. 20. AFBl ABT Mnu 

21. xd] xal ^oxo) x6 m. 22. ^illov ^. ^L"^. ^A. A"^ ^ 



328 CATOPTRICA. 

&6XE o^x &v(xxka6d^6ovrai aC BA^ B F o^sig iicl tb B 
oftfta. slg tb o(ifia 8i sl dvsxl&vro^ t6ai av at ymvCai 
TCQbg rotg A^ F iyCyvovro, 8si%%^6sraL 8i^ Tcav ixtbg 
rfig 7tSQLg)SQsCag yivrirav ro oftfia, rcc avrct 6vfi^aCvovta^ 
5 tovti^ri tb fi^ bQa6%^av tb oybyba 8i& tb tag ivaxXd6sig 
fii) ysvi^d-av iit^ aitd. 

^Ev totg xoCXovg ivdittQOig idv ixfiaXhv 8cdfistQ0V 
tr^g 6q)aCQag ix tov xivtQOv TtQbg dQd^dg dvaydyyg xal 

10 slg ro srsQOv fiiQog d^g ro ofifia^ oi8hv r&v iv rcaf 
a^rp iiiQSL^ iv cS ro Ofifia i6rCv^ dq)d^6srac^ rovri6tiv 
oiits r&v iTtl rfjg 8vayLsrQ0v ovrs r&v ixrbg rfjg 8va- 
[lirQOv. 

s6r(o xotlov ivoTtrQov ro AHd^ 8cdfisrQog 8h i6t(o 

16 ri]g 6(paCQag fj A^^ xal rfj A^ itQbg hQ^dg dvs6rdt(o 
ditb rov xivrQov rov Z ii ZF^ o^^a 8h i6r(o ro B^ 
o%l)ig 81 7] BE. ovxovv r] BE dvaxkcjfiivrj oix ^S^^ 
oiirs iitl ro B ovrs iitl ro Z* iv yaQ t^aig ycovCatg 
dvaxXarai. rii,SL aQa d)g fj E@. 6(ioCG}g 8s xal sdv 

20 ivrbg i^iti^ri ro o/Lt,fAa, ojrov ro 0, t) iitl rfig ^vafiirQov^ 
OTtov ro M, dvaxk(D^svaL at oiffSLg al SK^ MN f/lov- 
6lv hg at KA^ NlS. ovx ccQa bQaraL ov8\v r&v iv 

1. oipBig] om. m. 2. sig — bI\ sl yccQ slg tb ^inicc m. 

8i'\ scr. yaQ. av] om. VMv. ccl] om. M. ai — 3. 

iyiyvovto] iylyvovto cci TtQog toZg A, F 6r]iiBioig ycovlaL ' ovn bIgI 
oh tacct. o^bb' ccqcc ai BA, BF btpsig iTtl tb B bmicc icvccKXdiV- 
tcci m. 4. yiv7\tcci\ t£%'^ m. 5. tovtietiv Vv. Sloc — 

6. ccvt6\ vTtb ncc6&v t&v dcvocKXcD^ivcDV oijjscov sl iii} vnb fiovrig 
trjg diCi tov KSvtQov 7}yiisvrig m. 6. yivs^Q^cci M. 7. xs'] 

iy' Vv. 10. slg] mut. ixi ini "M.. 12. o^s (pr.)] oi;T« ti m. 

8icciLstQ0v'\ o-f 8 M, ut saepe. 15. %ai\ nsvtQov $1 tb Z, 

%ai ScTtb tov Z m. 16. Scno — Z] om. m. 17. 6'i^6ts v, 

, et V, sed corr. m. 2. 20. i\ins6si v, ts^^f^ m. 



CATOPTRICA. 



329 



si enim ad oculum refringantur, anguli ad A, F positi 
aequales fiant [prop. 2]. demonstrabimus autem, etiam 



r z 




si oculus extra ambitum sit, eadem adcidere, h. e. ut 
oculus non cematur, quia refractiones ad eum non 
fiunt. 

26. 

In speculis concauis si ducta diametro spkaerae e 
centro recta perpendicularis erigitur, et in altera parte 
oculus collocatur, nihil eorum, quae in eadem parte 
sunt, in qua oculus, cemetur, h. e. neque eorum, quae 
in diametro, neque quae extra eam sunt. 

sit AF^ speculum con- 



rJC 




cauum, diametrus autem 



sphaerae sit A^, et e cen- 
tro Z ad; A^ perpendicu- 
laris erigatur ZF, oculus 
autem sit Bj et BE radius. 
BE igitur refractus neque 
ad B neque ad Z ueniet; 
sub aequalibus enim angulis refringitur. itaque cadet 
ut E@. similiter etiam si oculus intra ceciderit in @ 
uel in diametro in M, radii ®K, MN refracti cadent 
ut KA, NS. ergo nihil eomm, cyiiae isl e^^iJftxssk. T^flsks^ 



330 CATOPTRICA. 

r^ ccvta iiBQSL^ ojtov i6tl tb oftftof, oiits t&v sicl trig 
dLUfistQOv o^ts t&v ixtbg tfjg 8ia[iBtQ0v. 

^Ev toig TCOLlovg ivdittQOig i&v iTcl tfjg SLafiitQOv 

5 tsd^ ta Sfifiata tdov &'jci%ovta tov xivtQOv^ oiditSQOv 
t&v 6ii(idt(ov 6(pd^6staL. 

s6t(o Tiolkov ivoTCtQOv tb AF^^ SiaiutQog d^b fj A/i^ 
xivtQov S^ tb Z, TCQbg ^Qd^&g 81 ^ ZF, Sfifiata 8s 
t& jB, E tdov aicixovta roi) xivtQOv^ o^ig Si ^ BF. 

10 oiKovv avaxlcofiivri ^^sl iicl tb E' iv t6acg yStQ yco- 
vCaig dvaxXataL. alXrj 81 ovSsfiia i]^SL avaxkcoiiivri 
aTcb tov B i%l tb E. st yicQ ^let (b^ ^ 5®, ins- 
^svx^^c^^av at ®E^ ®Z' 8L%a ccQa tiirjd^rl^staL ^ ifjcb 
B@E iTcb tfig Z0, xal avdloyov i6taL cog 4i BS 

16 TCQbg SE^ fi BZ TCQbg ZE' Stcsq aSvvatov ij ^lv y&Q 
BS (isit(ov i6tl tTjg SE^ ii 8s BZ t^rj ty ZE. oiSs- 
fiia ccQa 7]^SL dvaxlcoiiivrj dnb tov B iicl tb E. ^iLa 
ccQa oifLg ^6vov dvaxXa^d^ri^staL icp^ sxatiQOv tmv B^ E 
6^^dtG)v^ xal oix ^(pd^iq^staL tb E' oi) yaQ 6vfi7Cs6sitai 

10 fj Br ix^aXko^ivi^ tfj BA i%l td F^ A fJiiQrj^ icpaivsto 
Ss sxa6tov xatd tijv ^v^^okijv (i6vov t&v bQ(oyLivcov' 
01)8 s ii EF ov ^ii 6vyL7ci6ri t^ EA ijcl td F^ A iiiQrj' 
iv yaQ tolg xoCXoig iv67CtQ0ig sxa6tov tcbv bQCj^ivcov 
xatd tijv d%b tov oQO^ivov sig tb xivtQOv tfjg 6(paCQag 

25 dyo^ivrjv svd^stav bQatai. 

1. i6xiv Vt. o^xb] o^xs xi m. 3. %f'] 18' Vv. 6. 

xoL oiLiioixoi] xo b[niu M. 9. rof 5] e corr. M. xov] xov Z m. 

11. icvoc%Xai{LEvri — 12. rjh^sC] om. Mvm. 12. i]'] postea 

add. m. 14. B0EJB0E ycovia m. Z&] Z& s^b&siocg m. 

^GXocC} i6Xiv M. 15. @E] xr}v @E Mm. ZE] xijv ZE 

Mm. 16. i6xiv Vv. ZE] EZ M. 18. ^ovov} om. Mm. 




CATOPTRICA. 331 

siint, in qua oculus, cemetur neque eoruni; quae in 
diametro, neque quae extra eam sunt. 

27. 

In speculis concauis si in diametro ponuntur oculi 

aequaliter a centro distantes, neuter oculorum cemetur. 

sit AF^ speculum concauum, diametms autem 

ji^y centrum autem Z, et ZT perpendicularis, oculi 

autem B^ E a. centro aequaliter distantes, radius autem 

BF. refractus igitur ad E 
ueniet; sub aequalibus enim 
anguUs refringitur. sed 
nullus alius refractus a B 
ad E ueniet. nam si ueniat 
ut 50, ducantur @E, SZ. 
itaque LB@E a Z@ in duas partes aequales seca- 
bitur, et erit BS:@E=BZ:ZE] quod fieri non 
potest: nam 50 > @E et BZ = ZE. itaque nullus 
radius refractus a jB ad £J ueniet. unus igitur solus 
radius ad utrumque oculum 5, E refringetur, nec 
cernetur E. neque enim BF producta rectae B^ ad 
partes F, ^d uersus concurret, omnia autem, quae cer- 
nuntur, in concursu tantum adparebant [prop. 18]. 
nec EF rectae EA ad partes T, A uersus concurret; 
in speculis enim concauis omnia, quae cemuntur, se- 
cundum rectam ab eo, quod cemitur, ad centrum 
sphaerae ductam cernuntur. 



k^KocrsQov] 8cr. indrsQov. 20. (isqti'] "^ ® ^^^- ^* 21. 

itiaarov] ^yidrsQOv m. 22. EF] EF fnPaXXoiisvri m. cvy,- 
niasi Y, avfiTtsasZrai M. 



332 CATOPTRICA. 

jKi/ totg xodoLg ivdTttQOig i&v f^v ix tov xdvtQov 

Slxu tsfihv xal TCQbg dQd^&g &yayG)v dijg t& ofifiata 

t6ov ajtexovta tflg ix tov xsvtQOv^ d^fjg 8h t) &vd: (i66ov 

5 trlg SiafiitQOv xal tijg TtQog dQd^&g ij i^t^ aitfig tf^g 

XQog dQd^dg^ o^dSitSQOV t&v dfifidtcov q^aCvEtav. 

i6t(o xolkov IvoTttQOv tb AF^^ SidiietQog Sh ij ji^j 
ocivtQOv Sl tb K^ xal fi itQbg dQd^&g ^ KF Si%a ts- 
rfti}<y'9'o xatoi tb II^ TtQbg dQd^ag Sh aitri S6tco 'fj EIIZ^ 

10 Tial oiifiata ta 5, @ (ista^i) xsLfiei^a tfjg ts Sia^itQOv 
tfjg AA xal tfjg EZ iv TtaQaXXif^loig tatg EZ^ B0 
t6ov aitixovta tf^g KFj otlfcg Sh l6t<o 'fj BF &vaxXa}- 
liivri i%l tb &' t6ag y&Q %oist yoDvCag %Qbg tfl TtSQi- 
(pSQsCa Sl& tb TtaQalkrjlov slvav tf^v ZE rij B@ xal 

15 t6rjv tiiv BN tfi NS. xal iitilsvxd^st^ai aC KB^ 
K® ixps^k7]6d'(x)6av^ ix^sfilTl^d^tx) Sl xal ^ FB iitl 
ro O. xal iitsl ^sC^cov i6tlv 'fj BF tfjg BK^ ^sC^cov 
i6tlv r] P yovCa tfjg L &6ts xal ii V7tb FB® ybsCt,€ov 
tfjg vTtb SBK^ tovti6ti tfjg 'i)7tb B@K oix aQa 

20 6vfi7ts6sttaL 'fj Br t^ K®. oi)x aQa 6q>%"ri6stai tb 0* 
xat& yccQ tijv <yvftj3oAi)i/ (paCvstai t&v BFj K&. 

s6t(o Ttdhv ta avta r^ ijtdvo)^ ta Si 5, @ ^fifiata 
i6tG)6av iTtl tfjg SC^a xal TtQbg dQd^ag tsfivo^v^rig tiiv 

1. X7]'] Xfi' Vv. 3. 8i%oc\ ngbg dg&cig ovaccv rj $LafLSTQ(p 
8ixa m. icyocyiSov] dcyaymv svd^siav m. ral corr. ex rd 

m. 1 M. 4. L6ov] iiBta^v tfjg rs $iax^siar}g %at tov tiSvtQOv 
taov m. ^fjg — 5. dQd^dg] om. m. 6. TCQbg dQ&dg] ^ta^u-^f/- 
GTfg m. (paivstaC] (pavsttat m. 8. ij (pr.)] om. m. ij KF 
Sixoc] tfj Ad 7j KF %ai m. 9. %atd] 7] KF 8i%a yiatd m. 

TtQdg — }c6to}] yt,al dicc tov U SirJx^cD tjj KF TtQbg dQd^dg m. 

aifj] avtf]gYM.Y. 10. xf t/xfra] xftcy-O"© m, ^yjn-fW M. duc- 
{istQOv — 11. EZ (pr.)] EZ v,al tov K %svtQOv m. 13. l^ag] 
corr. ex lag m. 2 V. ^ 14. slvatl om. M. 15. BN] BH m. 



CATOPTRICA. 



333 




28. 

In speculis concauis si radio sphaerae in duas 
partes aequales secto et recta perpendiculari ducta 
oculi a radio sphaerae aequaliter distantes coUocantur^ 
siue inter diametrum et perpendicularem siue in ipsa 
perpendiculari coUocantur, neuter oculorum adparet. 

sit AFjd speculum concauum, diametrus autem 
A^y centrum autem K, et perpendicul^jis KF in 77 
in duas partes aequales secetur, et ad eam perpendi- 

cularis sit EllZy oculi au- 
tem Bj ® inter diametrum 
A^ et EZ in parallelis 
EZ,B® positi aequaliter a 
Kr distantes, radius autem 
sit jBT ad refractus; 
aequales enim angulos ad ambitum efficit^ quia ZE 
rectae BS parallela est, et BN = N®. et ductae 
KBy K& producantur, producatur autem etiam FB 
ad Q, et quoniam BF^ BKy erit LP^I» quare 
etiam LrB®> ®BK, K e. LrB®> BSK. itaque 
BF, K@ non concurrent. ergo & non cernetur; ad- 
paret enim in eo puncto, ubi BF, KS concurrunt 
[prop. 18]. 

rursus eadem sint, quae supra, et oculi 5, in ea 
recta sint, quae radium in duas partes aequales et 



r^] xifiv V. NS'\ G M, HG m. 16. iyipspX^ad^a)] i%' 

ps^XsLa&a) Y. xa/]om. M. PB] BrMm. .17 . fislttov {utr.)' 
listtov V. 18. P— 1] M FBS xfig iinb KB& m. rB& 
B@ M, BSrm. iistSov v, om. m. 19. @BK~\ BK M, 

KB@ m. TovrsaxLv V, comp. v. BSK"] KGB iisl^oiv 

iaxlv m. 22. X<s' Vv. fotrco] ^<?ra) di{ m. r||] xolg m. 

23. }^ax(oaav] om. M. xB\kvov^r^ vfyP^ tky^^^^i^ xtv^"^. 



334 CATOPTRICA. 

ix tov xdvtQOv inl tris AA. iittX ovv t^ri ^ ^ikv BF 
tfj BZ^ ii 8i r® tfi Z®^ TtaQciUrilog ctv sCrj ^ BF 
tfj ZS. oix &Qa 6vfiJCs6attaL fi BF ofcg tfj ix tov 
xBvtQov ijcl tb 6q6(1£vov^ tovti6ti r§ Z©, ijcl t& 0, F 
5 fiiQr^. &6t£ oi) (paCvatai tb ® oftfta* xata y&Q tijfv 
6vfi^okiiv iipaCv£to tcbv BF^ Z®. 

i6t(x) TcdXiv ta aitd^ tfjg dh 8i%otoii,Cag avc3t£Qa) 
x^C^d^cj tdc ofiiiata ta 5, F tdov aicixovta tf^g ix tov 
odvtQOv tfjg ZA. g^rifd di^ q)aCv£6^ai tk 5, F xal tk 

10 d^^ia &QL6t£Q& xal tcc aQt6t£Qic d^l^d xal tb ^tdcjXov 
li£t^ov tov 7CQo6(07Cov xal tb &7c66trjiia &jcb rov iv- 
oTCtQOv i%ov fi£t^ov tb £tSc3Xov. i6tco yaQ fj BA otJMg 
dvaxXcofiivri^ xal iic^^^vxd^co^av dTcb tov Z xivtQOV ixl 
td B^r at ZB^ Zr^ xal ix^£^lri6^Gi ^ BA. ijC£l ovv 

15 dvxotOfiCa i6tl ro iV, (i^C^cov i6tlv ii BZ tfjg BA xal 
r] K ycDvCa tf^g E. t6ri di ij K tfj ^' fi^C^ov &Qa 
xal r] /1 tfjg E. 6v(i7C£6ovvtaL &Qa at ZB^ FA ix- 
pXrjd^st^aL, 6v^7CL7Ctitc36av xatd tb 11. did td avtd 
dij xal al BA^ Z F 6v^7C£6ovvtaL xatd tb ®. dq^d^rj^^tac 

20 &Qa tb ^€v r iTcl toij 0, ro d^B iTcl rov 77, Tcal 
(paCvatai td ^iv d^^id dQL6t£Qd^ td S\ dQL6t£Qd S^i^La. 
dkXd [i^v xal fi£C^G)v ij SII tfjg BF' 7CaQdkkrjXoL yaQ 



1. iTci] om. m. tfjg] rd M. tcri] tari ^<^^^^ ni. 2. 

rg (utr.)] tfjg M. Post. ds del. t6 v. Z&] FZ m. 4. tovt- 
iativ V, comp. v. Z0] Z6>E M. 6. t&v] corr. ex ro 

m. 1 V. 7. X'9'' m, ^f Vv. 11. cc7t6] o icitBxBi th stScO' 

Xov m. 12. ^xov] corr. ex ^;^oi/ v, om. m. tb stSoiXov] tov 
ccTtoatrjiicctos, ov icnsxsi tb 7CQ66a)7tov m. 15. iati] iativ Vv. 
N] HM\m. ivslSov v. BZ] ZB Mm. 16. Jf (pr.)] 'bnb 
BAZ m. E] vTch BZA m. i) K {a.lt.) — 17. E] rg iikv 

vnh BAZ 7] ^Tth FAZ, ty 8s vTth BZA ij vnh FZA, Ui] &Qa 
7] vTth BAF oXrig tfjg vith BZF iLsi^wv iati m. 16. tff] corr. 



CATOPTRICA. 



335 




perpendiculariter secat, h. e. in A/1. iam quoniam 
Br=^BZ^ r® = Z®, Br et Z@ parallelae erunt. 

itaque radius BF non con- 
curret rectae e centro ad id, 
quod cemitur, ductae, h. e. 
Z0, ad partes 0, F uersus. 
ergo oculus ® non adparet; 
adparebat enim in eo puncto, 
ubi BFj Z@ concurrerent. 

rursus eadem sint, oculi autem 5, F supra punctum, 
ubi radius in duas partes aequales secatur, positi sint 
aequaliter a radio ZA distantes. dico, B et F ad~ 

parere, et partes dextras si- 
nistras, sinistras autem dextras, 
et imaginem facie maiorem 
maioremque a speculo distan- 
tiam habentem. sit enim BA 
radius refractus, et a centro 
Z ad 5, r ducantur ZB, 
ZFy et producatur BA, iam 
quoniam in iV in duas partes 
aequales secta est AZ, erit BZ> BA et iK>E. 
uerum L K = ^-^ quare etiam L^ > E. itaque ZjB, 
FA productae concurrent. concurrant in 11. eadem 
de causa etiam BAj ZF in. @ concurrent. ergo F 
in & cemetur, B autem in 77, et partes dextrae 
sinistrae adparent, sinistrae autem dextrae. iam uero 
etiam Sn> BF] sunt enim parallelae. ergo imago 




ex Sri M. iist^ov v. 20, tov (pr.)] corr. ex t6 M. 22. fwjfrl 

om. M. en] ne m. I 



336 CATOPTRICA. 

6l6Lv. tb &Qa eCdcolov g^aivstat fist^ov xal (ist^ov 
aTtixov rov ivdjttQOV iisl^cov yctQ fj MA tfi^ AA. 

iav 8b ^Ico trig SiaiiitQOV tsd^fj tk ofifiata^ t& d^l^ 
(palvBtai SE^ik xal t&. &Qt0tSQ& aQL6tBQii Tcal tb stSa)- 
5 kov ila66ov tov jCQ06(D7Cov xal iv t^ &vk fii6ov tov 
7tQo6(bjtov xal tov ivdTttQov. 

i6tG} yicQ ofifiata t& 5, F, xivtQOv Sh rb Z tov 
ivdTttQOv^ xal tfj SiafiitQp JtQbg dQd^i^g l6to fi AZA^ 
xal tavtrj TtQog dQd^o^g fj BF^ xal t6ri r§ BA i6tG} 

10 'fj AFj xal o^ig ^ BA avaxl(Ofiivri iitl tb F xal Sicc 
rov xivtQov aC BZKj FZE^ xal aitb t&v E^ K ii KE 
iTts^svxd^co. ovxovv tb (ihv B ijtl tov K (patvatat, tb 
Ss r iTtl tov E. ta &Qa Se^La Ss^lcc xal td^ aQL6xeQa 
&Qi6tBQa (paCvBtaL xal ro EK BtScokov ika66ov tov BF 

15 7tQo6(07tov' TtaQcillrilog yccQ i6tLV fi EK tfj BF' xal 
ccva ^b6ov tov iv67ttQov xal tov 7tQ06d)7tov (paCvBtaL 
tb BtScolov. 

avayo^Bvov Sh tov 7tQo6(x)7tov ItL lla66ov (paCvBtaL 

tb BtScoXov. B6tG) yaQ tb MN 7tQ66(*)7tov ro a^ro r© 

20 jBr* a(pB6trixbg a7tb tov BF XBCfiBvov bfioCog. oi^xoiyv 



1. sioL Mm. 2. iisi^ov v. 3. X' m, Xri' Vv. 4. 

de^Lcc (pccLvetaL m. 5. ^Xaxtov M. ^Liaov^ iisGa) Vv. 

7. tov ivdjttQov tb Z m. S. AZd'] AJ Y^ aei corr. 

m. 1. 9. tjj BA 1671 m. 11. BZK] BZE M. &7c6 

— KE] om. m. iTtsSsvx^co ij EK m. 12. B] F m. 

tov] corr. ex to M. K] E M. 13. F] B m. %ai] 

(focivetca %ai m. 14. q>aivBtai] om. m. 18. JeXat- 

tov M. 20. B r (alt.)] B F %ai m. oi^ovv] d^ovv V, corr. 
m. 1. 



CATOPTRICA. 



maior adparet^ et eadem magis distans a speculo; 
nam MA > AA, 

sin extra diametrum ponuntur oculi, partes dextrae 
dextrae adparent^ sinistrae autem sinistrae^ et imago 
facie minor interque faciem speculumque posita. 

oculi enim sint 5, F, specuii autem centrum Z, 
et ad diametrum perpendicularis sit AZA^ ad eam 




autem perpendicularis jBJ) et sit BA = AF^ B^ autem 
radius ad F refractus, et BZK^ FZE per centrum 
ductae, ab E^ K autem ducatur KE. itaque B m K^ 
r autem in E adparet. ergo partes dextrae dextrae 
et sinistrae sinistrae adparent, imago autem EK facie 
BF minor; EK enim rectae BF parallela est. et 
inter speculum faciemque imago aA^^x^\». 

Suclidea, edd. Heiberg et Menge. "VXl. ^^ 



338 CATOPTRICA. 

71 ccTth xov M ijtl ro Z xevzqov i7tLt,Ev%d-ai6a xal i%- 
^hfi^Eida avmtSQOV 7Cs6eitaL tov K cag tb A^ 'fj Sh aith 
tov N STtl ro Z avatSQOv tov E d)g th S, (paCvstai 
aQa th MN &g ro @A. xaC s6ttv ska66ov xh &A 
5 rov EK xal syyiov tov ivdTttQov. 

Xd'\ 

^vvatdv i6tiv ^voTttQOv xata^xsva^d^fivat S)6ts iv 

rc3 ait^ g^aCvs^d^at itksCcD 7tQ66G)7ta^ ta (ihv fisC^ova^ 

ta Sh ika66ovaj xal ta ^hv syyiov^ ta Ss 7tOQQ(DtSQOv^ 

10 xal tG)v [ihv ta Ss^lcc Ss^tdj tic S\ aQL6tSQa aQL6tSQdy 

tG)v Sh ta aQL6tSQa Ss^td^ ta Sh Ss^Ldc aQL^tsgd, 

i'6t(o yccQ ijtCTtsSov ro AM. ovxovv iv rovrp 
ysvoLt^ av xvQta fisv svoTttQa oia tdc AOF^ GPK^ 
xolla Ss ola td F^E^ ZH®^ i%C%sSa S\ ola ta EZj 



A M 




l^ AM. tsd^svtog ovv tov 7tQo6G)7iov^ oTtov ro iV, (paCvstaL 
dith iisv t&v iTtLTCsScov C6a td slScj^a xal l6ov dit- 
s%ovta^ dith Ss tcbv xvgt&v ikd66ova xal sXa66ov 
djtsxovta^ dith Ss tcbv xoCkov itavxoSaTtcbg^ xad^djtSQ 
SsSsLXtaL. 



2. wg t6] ^'(og tov m. A] om. M lac. rel. 3. N] 

E M. ag] iag VMvm. ro (alt.)] V, et v seq. ras.; 



CATOPTRICA. 339 

facie autem retracta etiam minor imago adparet. 
sit enim MN eadem facies ac BFy sed sl BF remota, 
et similiter posita. recta igitur ab M ad centrum Z 
ducta producta supra K cadet uelut in ji^ recta autem 
ab iV ad Z ducta supra E uelut in @. itaque MN 
ut ®A adparet, et GA < EK speculoque propior. 

29. 

Fieri potest, ut speculum construatur eius modi, 
ut in eodem complures facies adpareant, aliae maiores, 
aliae minores, et aliae propiores, aliae longinquiores, 
et aliarum partes dextrae dextrae, sinistrae autem 
sinistrae, aliarum partes sinistrae dextrae, dextrae 
autem sinistrae. 

sit AM planum. in eo igitur construi possunt 
conuexa specula ut AOFy SPK, concaua autem ut 
FAEy ZH&, plana autem ut EZ, AM. facie igitur 
in N posita in planis speculis imagines aequales 
aequaliterque distantes adparent [prop. 19], in conuexis 
autem minores minusque distantes [propp. 20 — 21], in 
concauis autem uarie, ut demonstratum est [propp. 
24-28]. 



xov Mm. &] @. xal i7tsSsvx^(o ij @A m. 4. ro (pr.) — SA] 
ta M, N Tiocta tcc @, A m. ^Xattov M. o. EK] EM M. 

6. x^'] Z^' Vv, Xa' m. 8. TtQOGmnanf M. 10. t&v] m, 

roS Vv, td M. td (pr.)] om. M. ta ds] t& Ss td Vv. 

11. t&v — ScQLatsgd (alt.)] om. VMv. dQiatSQa Ss^id] Ss^ta 
dQLGtSQd m. ds^La &QL6tSQd] &QL6tSQa ds^Ld m. 12. oi^K' 

ovv] %ai Mm. 13. ysvoLt &v] ysvs^&o) m. yivQtd] nolXa m. 

14. TiotXa] nvQtd m. td (pr.)] supra scr. m. 16. AM] 

KM m. oTtov t6] avad^sv tov m. N] H V, if Mvm. 

16. sHdoiXa'] l'da)Xa V. 17. iXd660va] iXdttova M. ^Xa66ov] 
^Xattov M. 18. 'nad^dnsQ] xal nad^dnsQ M. 



340 CATOPTRICA. 

^Ex ratv tcoHcdv ivdntQcov TtQog rbv 7]hov rsd^avrc^v 
7CVQ i^ccTtrerat. 

itSrco xotkov ivoTCrQOV rb ABF^ ^Atog d\ 6 EZ^ 
5 xivrQOv d\ rov xardytrQOv ro 0, xal &7t6 rivog 6rjiiSL0v 
roi) ^ iTCi^svxd^sttSa [ihv inl ro @ xivrQov ii jdS ix- 
Ps^lT^tSd^cj ijcl ro 5, 7CQ067CS7trc(ixirc3 8\ ii ^F dxtlg 
xat avaxsTcld^d^cj i7cl ro K. dvaxla^d^i^j^srat dij i7cdvc3 
rov S xivrQov 'fj ydg yovCa fj 7CQbg r^ 7CSQiq)SQSicc 

10 '^ n ild66c3v i6rl rr^g TCQbg rfj TCSQicpSQSia koi7Cfig rijg 
i)7cb BF^, xal s6rG} ii AB 7CSQi<psQSia i'6rj r^ BF^ 
xal d7cb rov /1 allrj ng dxrlg 7CQ067Ci7crircj ^ ^A. 
q)avsQbv ovv^ Zn dvaxkcofiivrj fj A^ dxrlg 7Cs6slrai 
i7cl ro K Sid ro t^rjv slvai rijv AB TCSQifpsQSiav rg 

15 ^jT. 6[ioiC3g dh Ssii^^ilfisrai^ on 7ca6ai aC dTcb rov A 
7CQo67Ci7crov6ai TCQbg ro svoTCtQOv xal l'6ag d^colaa^d- 
vov6ai sig ro avtb 6vfi7Cs6ovvtai tfj BS dvcotSQOv 
tov 0. 

^6tc3 Tcdliv xoilov svoTCtQov tb ABFj ijXiog di 6 
20 AEZ^ xal dTCO tivog 6ri[isi0v tov E Sid tov xivtQOv 
^6tc3 fi E®B^ xal dTc" aUc3v [Sio] tcbv z/, Z ai A0^ 
Z SA. ovxovv 7CQoSsSsi%a^sv^ oti at d7cb tov E dxttvsg 
6v^7ts6dvvtai slg savtdg Sid tdg II^ P ycovCag l'6ag 
ov6ag' Sid^stQOi yaQ sC6iv' aC Ss dTcb tov Z Sid tdg 



1. X'] ft' Vv, X§' m. 7. TtQoansTtTcoyisto v. dF] 

jrx V. 8. dri] $i M. 10. ildtroiv M. iaiiv Vv. 

tfjg (pr.)] tjj V. tfjg Xomfig tf]g V. IS. AJ] dA m. 

nsasitaC] TtQoaTtsasttca M. 15. diioicog] /^ V. 16. nQog] 
ccnttvsg TtQog m. taag nsQLtpsQsiag &7toXaiL^dvovaai i%atSQa)- 
^sv tov B m. 17. avt6] om. M lac. rel. 19. /lmx' Vv, 



CATOPTRICA. 



341 



30. 

E specuKs concauis aduersus solem conuersis ignis 
adcenditur. 

sit ABF speculum concauum, EZ autem sol, et 
centrum speculi 0, et a puncto aliquo ^ ad centrum ® 

ducta z/@ ad 5 producatur, 
adcidat autem z/F radius et 
ad X refringatur. refringetur 
igitur supra centrum @; nam 
/. il ad ambitum positus mi- 
nor est reliquo angulo ad 
ambitum posito, scilicet BF^, 
et sit arcus AB arcui BF 
aequalis^ et a z/ alius radius 
adcidat ^A, manifestum est 
igitur, radium A^ refractum 
ad K cadere, quia arcus AB arcui BF aequalis est- 
eodem modo demonstrabimus^ omnes radios a ^ ad 
speculum adcidentes et aequales arcus abscixidentes ui 
eodem puncto supra @ posito cum recta B0 eojocurrere. 

rursus ABF speculum concanum «it, 8ol autem 
jdEZy et a puncto aliquo E per @ eentruju ducta 
sit E@B, ab aliis autem z/, Z rectae /d^S\ Z^A. 
antea igitur demonstrauimus, radioB ab E ducto^ per 
se ipsos refringi, quia iTI^P fprop. -^JJ: diainetn 




ly' m. 21. &XX(ov\ -nyif e coir. v dut, oir. « 

itf* M. rag] -^ e corr. V, -rf/t JW ^ 

bIciv\ om. m. 



«&. «kj 




342 CATOPTRICA. 

Kj A ycoviag, «f di &7ch rov A stcI tiiv AF Si& tag 
Nj tS ycoviag l!(Sag ov6ag. Ztv 8\ %a(Sai avtal eig 
aavt&g avaytk&vtai^ 8i]Xov* ix tov yccQ zavtQOV ov6ac 
rj^ixvKlta 7Coiov6lv^ aC 8a tG)v tj^ltcvxIlcov ycoviai i!6ai 
5 si6iv' dfc' H6C3V aQa ycovicov ai &va7ika6evg yiyvovtai' 
eig eavt&g ovv &vaKlG)vtai, Jta6av aQa 6vii7te6ovvtav 
ajcb Tcdvtcjv t&v ^rjfieicov ijcl tag 8va rov xivtQOv xal 
iv rc5 xivtQC) [^axttvag^. roxJroi/ o{>v t&v dxtivov 
ixd^eQiiavvoiiivcov TCe^l tb xivtQov tcvq ad^Qoi^etav. &6te 
10 ivtavd^a 6tv7C7Cvov ted^ev i^acpd^i^^etav. 



1. K, A] TtQog xm A m. yoiviag] ywviccg taocg o^aocs oiioLag 
ccXXrjXocig m. inl tijv J F] om. m. 2. N, ISI] ngbg tc5 F m. 

Heag o^aag] o^aag Haocg M, Sid^L^tQoi ydq slai n&aoci m. 3. 
iv. Tov yccQ] 8x yccQ tov M, 8icc yccQ zov m. o^oaoci] lovaai m. 

4. rjnLTivKXLa] rj^iKVHXiov M. noio^ai M. r&v Tjfiiv.vnXlav] 
To5 rjiiLTiviiXico M. yoDviai] yaviai al yiv6nsvai Ttgbg rotg ns- 
Qaat T&v Sia^stQiov xal TtSQisxo^svai vit' aiyt&v ts t&v Sva- 
iistQav 'nal t&v 7tSQiq>SQSL&v m. 5. siai Mm. yivovtai M, 
yivovtai v.al dicc tovto m. 6. ovv] om. m. 8. Scntivag] 

deleo. 10. atvTtJtiov] v.al vTtmov M, supra ser. m. 2: <5V7ti,ov. 

In fine: BJijnXsidov v.atoittQi^d V, tsXog m. 



CATOPTRICA. 



343 



enim sunt; radios autem a Z ductos, quia i K = A, 
et radios a ^ ductos per ^Fj quia i N = S. omnes 

autem per se ipsos refringi, 
manifestum est; nam cum e 
centro ducti sint, semicirculos 
efficiunt, et aoguli semicircu- 
lorum aequales sunt; sub aequa- 
libus igitur angulis refractiones 
fiunt; itaque per se ipsos re- 
fringuntur [prop. 8]. omnes 
igitur radii ab omnibus punctis 
in radios per centrum ductos 
et in centro cadent. his igitur 
radiis incalescentibus in centro 
ignis colligitur. ergo stuppa 
ibi posita adcendetur. 




jifii 



m 



!. VI 



it 






I 



SCHOLIA m CATOPTRICA. 



I- 






1. &8(OQOviievov rtvbs ii^ovg p. 286, 4] t) Tcdhv 
itBQov tLvbg 6(oiiatog TCQbg d^d^S^g ycaviag i6tafiBvov 

rw BTCLTCBSp^ BV S Xal tb ivOTttQOV XBLtaL 

2. Tfjg 6q}aiQag p. 286, 15] siTtB Si tb xbvtqov 
tfjg dfpaiQag xal ov ro xbvtqov tov ivoittQov^ iTCBiSij 6 
tSq^aiQOBiSig i6ti tb xvQtbv ivontQov. &67tBQ oiv ijtl 
trig 6(paiQag B%Bi^ oti^ Sd^Bv av voTJey tig ijt' aitrjv 
ixfiakkdiiBvdv ti fidcQog djtb tfig im^pavBiag trjg 6(paiQag^ 
ixBivo tb paQog Sid tov xivtQov ikBii^Btai* vBv6Bi ydQ 
dsl ipv6ix&g TtQbg tb iii^ov^ xad^d xal ro5 ®BoSo6i(p 10 
djtoSiSBixtai iv totg U^paiQixotg' ovt(o di) xal iitl tov 
6(paiQOBiSovg ivdittQov idv JtQbg dQd^dg ycovCag dnd 
tivog oQiDiiivov d^pBd^fl tig Bvd^Bta^ jtQbg tb ocivtQov 
tov ivdntQOv jtB^Bttai. 

3. Oifxiti bQatai p. 286, 12] o^bxovv iv totg im- 15 
TtiSoig ivdjttQOig Bxa6tov t&v dQCjfiivcov bQcctai xar' 
ixBtvo tb fiiQogj xad"' 8 6vii7ttcj6ig yCvBtai ixpallo- 
fiBvcov tijg tB SipBcjg xal trjg djtb tov bQcoiiivov xad^- 
itov^ &6jtBQ ijtl tov iTtoxBiiiivov vjtoSBCyiiatog ro ^ 

1. p. 2. V*p. 3. Vpq. 



7. avti/jv] p, ccvt6v V. 8. tC] p, om. V. 9. iKsZvo] V, 
iyist p. vsvcsl] dubium et in V et in p. 12. 6(pccLQosL$ovg] 
V, ecpaLQLtio^ p. 16. 6%6Xlov add. m. 2 V. 19. i^nov.snCsviyv] 
u. nr. 4. -J T(J] Vq, 8s p. 




348 SCHOLIA IN CATOPTRICA. 

r6 dQafisvov l^co Soxat l6G) dQcc^d^aL iv t& i(S67CrQC3 
7cat& tiiv 6viL7ttco6iv, 

4. Tov E xataXrjq^d^dvrog 
oiTcitt bQatai to 6q(o[18vov^ 

5 8 xata [lav tb dlrjd^lg 6^(o 
bQatai ro z/, Sotcovv S^ bQa- 
(Sd^ac TCQbg tfj 6vii7tt(b6£L, Va 

5. Tov E Tcatalfj^pd^svtog (ydxstt bQatav ro 6^c6- 
lisvov^ icftt tb id^ xatct fiiv tb dkr^d^sg bQatai 

10 TtQbg tS tdnc) rp avtiXQi) tov 5, (paivdfisvov Sh TtQbg 

tfj 6Vll7tt(h6£L. 

6. OvKstv bQatai p. 286, 15] ovxovv iv totg xvq- 
totg ivdntQovg sxa^tov t&v bQCjfisvcjv bQcctat Tcat^ 
ixstvo tb [iSQog^ xa-d*' o ^iigiTCtcj^ig yCvstai ixfiaXko- 

15 ^svcov tijg ts oxljscjg xal trjg ajtb tov bQcoiisvov i%l 
ro KsvtQOV ijtit,svyvv^svrig svd^svag, 

7. 'O Ss ^AQiLiiriSrig otirca ksysv^ otv fi Z ycjvva 
tfi E r] l6ri i6tlv rj ildttcov r) fisv^cov, s6ta) %Q6tSQ0V 
fisv^cov 7] Z trjg E' ildttov ccQa ri E. iTtoxsv^d^cj ovv 

20 Ttdlvv a^gia tb z/, xal ajtb tov b^^atog itAXvv ava- 
Tisxld^^co iitl tb bQCJiisvov tb B. i6tav ccQa fj E ycovva 
gisv^a)v rfig Z. rjv Ss xal iXdrrGjv 57tSQ aroTtov. 

8. '^AAa xal r] @ r/J A p. 288, 16] tJ ort r] xsQa- 
rosvSrig ycDvva d7td6r]g ol^svag ycovvag iXdrrcov i6rvv^ iq 



4. V in mg. inf., pq. 5. V in mg. inf., pq. 6. Vpq. 
7. Vp (ad prop. 1) (q). 8. Vp(q). 



5. o] om. q. 12. gxoXiov p, m. 2V. 18. ri (pr.)] m. 2V. 
ri (sec.)] m. 2. V. i) fta/Joi/] m. 2 V. 19. ftftfoi; V. 

24:. SisKxg] d^iag postea ins. m. 1 V, om. p. 



SCHOLIA IN CATOPTRICA. 349 

iav anb tov kbvtqov iTti^s^d^cofiBV ijtl tiiv krpr^v^ Slrj 
tfl VTcb jK, ji i'6rj eOtai 'fj tov ii^LvxvKkCov tf] tov fiybi- 
xvxkcov l67i i^paQiio^ofiivov. Xovxii «pa ij ® tfj ji L'6rj, 

9. ^Avi^ovQ p. 290, 17] f^yovv 6i,slav xal &iiL^kBlav^ 
07CBQ yCvBtai TtlayCcog Bl6^akkov6rig tY^g ixttvog. l 

10. 'EtcbI ovv fj ® trjg M fiBC^mv^ xotvri xqo6- 
xBCad^co fi K^ A, 8<>o aQa aC @, K^ A Stjo tcbv K^ A^ M 
(iBC^ovg. ai Si K^ A, M dvo dQd^atg taai' aC @, K^ A 
&Qa S^do dQd^&v [iBC^ovg. tkg Ss ajt' iXattdvov ^ 

SvstV dQd^&V 6V^7CC7CtBVV. IC 

11. I}%6kiov, iiCBiSii yccQ^ o6rj i6tlv ri cctco tov 
^(ifiatog iTcl tb SvoictQov svd^sta^ to6avtri i6tl xal fj 
avtavaxkcjiiBvri anb rov iv6ictQov TCQog i'6ag ycovCag 
avtri Sta tbv Sqov^ S6ti Sid: tovto fj [ihv BF tf} Fjd 
i'6r]j fj Sh BA tfj AE^ ijCBiSij ro Sftfta jCQbg t& B if 
i6tLV. avi6og Ss ij BF tf] BA' avi6og aQa Tcal ri FA 
tfl AE. oix ccQa 6vfi7CB6ovvtaL Sia tovto Si& tb tijv 
lisv (isC^ova slvaL^ tijv Ss iXattova. oiSi ii^ietav 
avifffiaL tijv FA xal ayaystv scjg tov E' to6avtri yicQ 
slvat dcpsCXsLj 86r]7CSQ xal rj BF sifd^Bta 'fj axtCg^ to- 2( 
6avtrig S\ aitrjg v7C0XSLiiivrjg 7CQbg trjv AE ov ysv^^staL 

6V[l7CtC36Lg. 

12. '76aL ccQa b16Cv p. 294, 17] xatic tb iq^aQfi^^s^d^aL 
tcc ijfiLXvxlLa. 



9. V^p. 10. V (ad prop. 4 part. pr.) (q). 11. V^ (ad 
eandem). 12. Vq^ 



2. Post ^Gvai deest i} K, @. &XX'. 5. nXayiag] V, Ttdv- 
roff p. 6. &] e corr. m. 2 V. 11. S<yrj] tari V? 16. Post 
iativ del. (isiioav Sl 7} BA Y. 18. Ajite o/d^i ^A.^. wsjr\^ . 



350 



SCHOLIA IN CATOPTRICA. 




13. Msi^cov 'fj Z ymvia p. 296, 4] iTCBvdri Tcavrbg 

Tcvxkov rii7]iiurog aC YcovCai t6aL sCtSLV olov r[jL7l[iarog 

rov ABF iav rsfLVCDiiav 8L%a rijv AB olov xara rb N 

xal TtQbg OQ&ag dvaerTl^cofiev 
6 rijv NFj iq)aQii6iov6LV al iCQbg 

rotg Z, B ycavCaL^ xal xard rbv 

r&v iq)aQ(io^6vrc9v l6yov xal 

l'6aL 86ovraLj iTCBLdii xal ro FNB 

iq)aQ^6iBL rip FNA. S^d rd avrd xal aC rov FB 
10 r[i7]fiarog ycovCaL CtSaL sCdCv. iTtsl ovv iibC^cov i6rlv ij 

vTcb ZBF yc3vCa rTjg vicb FBP^ ^bC^cjv xal 'fj Z rrjg 0' 

C6ri ydg 7] (liv Z rr] hnb ZBF ycovCa^ i5 ^^ ^^^ FBP 

rri TCQbg r& F, xal ravra [ilv d}g i%l rov Qrjrov, orL 

Sh xad^^Xov fi rov fiBC^ovog rii7]fiarog ycjvCa olov hg 
15 ii vTcb FAZ ^bC^cjv k6rLV rr^g rov ikdrrovog r^^liiarog 

ycjvCag rrjg vicb EZS^ SBi%coiiBV ovrcog' iercj ydQ fi 

vTtoxBLgiBvrj xarayQa<pi] xivrQOv 

ovrog rov H, iiCBi ovv aC rCbv 

ri^LXvxUcov ycovCaL i'6ai Bi6lv 
20 xard rbv rcov i(paQ^o^6vrc()v 

k6yov^ l6ri rj i)7tb KA@ rfi 

vTcb AZ@^ G)v ri {)%b KAF 

ikdrrcDV i6rl rijg vTcb AZE' 

ijcl ildrrovog ydQ TtBQiq^BQBCag 
25 pB^rjXBv rrjg FK' XoiTcij ccQa i] 

vjcb FAS ^bC^cov i6rl rrjg ijtb EZ&, OTtov yCvBrat 

ydQ ro ikarrov^ ixBl ro [iBt^ov, o). B6ri Sh avr^d^Bv 




13. V(q). Fig. pr. om. V. 



3. rrjv'] Tov V. 6. xat] om. V. 26. yivst ai] -f V. 

27. o)] h. e. OTtsq i6u 5er|at. ^6ti\ sqq. om. q. 



SCHOLIA IN CATOPTRICA. 351 

ix tov iv t& y' Ei^TclsiSov' iv xvxkfp fj ^lv iv rp 
'fj[itxvxkLG) xal ta i^ijg [III, 31]. 

14. Tov yaQ ^Si^ovog t[i7](iatog fj ycovia. xal itdUv 
iicv trjv aTtb rov xavtQOv iTtL^av^cofiev i%l tdc jT, A^ 
xatcc ta cf^rii i^tat. l 

15. AC aQa Z, H p. 296, 5]^) iccv yccQ a%o tov 
xavtQOv ijtl ro A xal F im^av^c^fiav^ al ycvdfiavaL 
TtQog tm A t&v fjfiLxvxkmv dvo ycovLai^ tovta6tLV at 
y a(ia at H^ A^ Z, tatg yLVO^iavaig TtQog rc5 F t&v 
il\LLXvxXLCov Svo ycjvCaig^ tovta6tLV tatg tQL0lv a[ia 1( 
tatg K^ M^ 0^ t6aL al6LV' hv at JF/, Z [iSL^ovag iSsLxd"ri- 
6av tcbv jfiT, S' loLTtij ccQa fj A XoLTtflg trjg M iXattfov 
i(StLV' STtov yccQ tb (lat^ov^ ixat tb iXattov, 

16. 'H Sa n trig O p. 298, 7] iicv iTtLlailmiiav ccTtb 
tov xavtQOv iitl ta A^ f, &)g iv ta 6%okL(p tov TtQb 11 
ai)tov [15]. 

17. ^avaQbv Sa p. 298, 13] i%al yccQ i'6rjg ov^rjg 
trjg A® tfi FK tarj iSaCx^rj xal fj 11 tfj O^ iiaL^ovog 
Sh ov6rjg tfjg A ® tfjg FK ikd66c3v iSaix^rj rj O tfjg U^ 
idv fj 6v[i7tta}6Lg iTtl tfjg itaQLcpaQaCag yivrjtaL hg xatd 2( 
ro 2:, t6ri a6taL f} H® trj 2JK r) ildttcjv fj Z® tfjg 



1) Huc refertur in Vq, sed pertinet ad XotTtri &qoc 
p. 296, 7. 



14. V(q»). 15. V(q). 16. V(q^). 17. V(q). 



1. 1^] corr. ex si m. 2 V. iv (tert.)] s V. 3. iisliavos 
V. 4. KSvtQov] LS Y, s? m. 2. 7. ysivo^svai V. 9. 

A\ A V. ysivo[LSvai$ V. 



352 



SCHOLIA IN CATOPTRICA. 



2JK' aC yaQ ycavCai tbv aiftbv loyov i%ov6vv tatg 
TtSQKpsQsiatg^ dig iv t& ^' t&v 6toi%BL(ov [33]* oxbq 

A 






aSvvatov. 7toXk& d^ %kiov (yb8\ ixtbg 6vii7tB6ovvtaL 
C3g iTcl tfjg /3' xatayQaq^fjg' tcoXX^ y^Q ro &Svvatov, 

5 18. Avtri fj xatayQaqyfi ov Tcatcc t& ajtodBix^BVta 
i6tlv ixPaklo^iBvcDv t&v 8^£oi/ xal tav 6qc3[ibvg}Vj 
o^dds fj tov pcpUoVj dXXi^ avtr^ Tcatd tb iv totg OQOig 
ix^aXXoiiivov twv Sifscov xal xad^itov ayofiivcov aTtb 
t&v bQca^ivcjv xal ixpakkoiiivcov ^ xad^b yj 6vgi7ttco6Lg 
10 yCvstaL^ bQco^ivcov twv bQtofiivcov. 

19. ^vvatbv xaC^ cog sxsl rj xatayQatpr^^ TtQo^rlvaL 
tijv dst^LV, iTCSLSrj y^Q avcjtiQa i6tlv i^ BA trjgBr' 
idv ydQ avc3&sv S^d ait&v dydycjiisv xdd^stov^ td xatd 
tijg BA td dv(DtSQd i6tL tflg xad^itov tb aTtb tfjg 

15 dvcjtiQag 6Q(biisvov^ 8 iatL tb z/, tb dv(htsQ6v i6tLV, 

20. Kal tovto b^oCcjg tm dvcoti^av slvaL f^v BA 
tfjg BF. idv Sh xatd tb iv totg OQOLg ixl twv kvq- 
t&v^ ots sxa6tov t&v OQcofiivcjv iv avtotg bQcctaL^ 



18. V(q); ad prop. 9. 19. V(q); ad schol. nr. 18. 20. 
V(q); ad prop. 10. 



4. ccdvvatov] sc. ^sl^ov. 7. ScXXcc] om. q. 8. i-nPccXXo- 
^svav] q, ifiaXXoiisvav V. 12. ScvoaTSQo] icvwxSQ V, &vtoriQto q. 
14. icvihtSQa] ScvcotSQ V, ccvatSQa} q. 



SCHOLIA IN CATOPTRICA. 353 

ocad^b '^ (fiifiJttai^Lg yCvBtai ixPaXXofidvmv tfjg S^fcog 
ocal tfjg hcl tb ocivtQOif im^vywiidvfjg^ &Xl€3g i6tac 
'^ ocatayQaqyi^' dfiOLCog xal iicl t&v akk(ov. 

21. Katii ro <i%6Uov tb iv tp a [8]. 

22. '0(pdii(fstat aQa i%' svd^siag p. 314, 5] iocsLdii 6 
tb A a^btb oi)% bQatav iv t& itfdntQpj &kla tb si^Sco- 
kov airtovj 8 foco icov tfj vo7^6st tov i^dxtQOv 6q(o- 
^isvov xatd: tijv (fvfiTttmtfLv dQ&tat xatk tb E^ iocsLdi^^ 

sl iic^ sv^sCag sl6lv aC bQa^sig^ tb B l66) nov tov 
ivdTCtQOv oijfstai^ sl Ss i6o^ icvayxri &xb tov bQmfiivov 10 
i^d^st^av slg 6viiactGi6vv aircrjg (pd^d^ai^ Sg i6tai tdTCog 
tov i66) Soxoihnog iv tp ivdTCtQO} q>aCvs6^aL. 

23. Kal insl t^rj i6tCv p. 316, 9] Sta tb t&g ^v 
&vaxl(o(iivag t6ag slvai^ ixfikrjd^sC^rjg Sh tfjg ®r t&g 
xata xoQvg>iiv t6ag slvai, 16 

24. Kal t6ov tb bQofisvov p. 316, .14] i&v i%i- 
%sv^GilLSv &7cb tov K xal E inl ro 0, Svo at KZ0 
Sv6lv talg EZ® t6ai^ xal ytovCa xal ycavCa^ &6ts xal 
yfovCa fi inb K0Z t6ri, insl ovv SXri ii iicb A@Z 
olfj t^ {fTcb ^®Z t6ri' bQ%ii y&Q sxatiQa' i^ &v ai 20 
7CQ0Qri^sl6av t6ai^ koiicii aQa ^ 'bnb E0A XovTCfl tfi 
{)7cb K@A t6ri, insl oiv Svo at K@A Sv6lv talg 
E@A t6av^ Tcal ycovCa ycjvCa^ xal P&6vg ij AK pd6st 
tri AE t6ri. 



21. Vq»; ad p. 312, 2. 22. V(q). 28. V(q^). 24. 

T(q). 



9. ^fffi)] l^ao V. 18. xa/(8ec.)] g V; fort. tari. l^. K&7A 

scr. KBZ t^ 4>nb E&Z. 20. ^9Z] J supra scr. m. 1 V. 
23. taaC] tgrj V. 

Xaolide§, edd. Heiberg et Menge. TQ. "^ 



354 



SCHOLIA IN CATOPTRICA. 



25. ^('X^ «V ^H tstiirjiievi] p. 318, 10] iTcel ohf 
ai Siii tijv &vdxka6LV t6ai^ ii, &v al a%oXa\L^av6iiiBvai 
TCQog tfi neQiq>8Qsla ino r^g KP t6ai Si& ro 6%6kiov 
to iv t& a [8], Xoiitii aqa i\ i)%o Pj4B Xolxtj tfj r)7cb 

h KAO l'6ri. akk" fi iTcb PAB tfi ixb ZAK terj ietiv 
xata xoQvq)iiv yaQ* xal i] i)7cb OAK &Qa t^rj tf^ imb 
KAZ. 

26. MeCl(ov aQa ij EK p. 318, 11] ictio^ tQiyGivov 
tb AZE^ xal 8C%a tetii^^ed^co ij A ycovCa t^ AK^ Ttal 

10 ietco afifikeLa ii i)%b AKE. X,eym^ Stt [lei^ayv i6tlv 

il EK tfjg KZ. iix^co yccQ 

xdd^etog i] AA^ xal TceQl tb 

tQCycyvov tb AZE xvxkog 

7ceQiyeyQa(p^(o^ xal ixfie^Xif\- 
15 6%^(o6av ii AA xal ij AK. 

ehe S\ 6i,eLa etr^ ii Z elte 

^Q^^ii ehe d^^pieta^ TCQO^aCvei 

ij djc6SeL^Lg. ei yaQ t6ri e6tLV 

il vTcb KAE tfi i)%b KAZ^ 
20 l6ri xal ij EN TceQLtpsQeLa' 

^eC^cov ccQa ij EM trjg MZ. ^st^ov aQa xal tb 

dnb ME tov dxb MZ^ tovts6tL td dnb MA^ AE 

tcbv dTcb MA^ AZ. xoLvbv dfprjQT^^d^a) tb dnb MA' 

XoLTcbv ccQa tb dnb EA tov dnb AZ ^sl^6v i6tLV. 
25 KOLvbv TCQO^xsC^d^G) tb «JTo AA' td aQa dicb EA^ AA^ 

25. V(q). 26. V(q). 




2. /Wt] yiai V. dcTtoXcciipccvoiisvciL'] dcjtoXccii^avoiisvav ccl V. 

4. Post T(J ras. 1 litt. V. 6. •iiOQvcprjv] 7iOQV(p^ V. 7. 

KAZ] e corr. q, KAZ V. 9. tsriiriad^a)] rsccx^riad^o} V. 18. 

ei] scr. iitsi? 20. 7} EN] EN V. Post 7tSQi(psQSLcc ad- 
dendum tjj MZ 7tSQL(pSQsicc. 



SCHOLIA IN CATOPTRICA. 355 

tovTBtftL tb €atb AE^ inBitfiva twv &%o ZA^ AAy tovt- 
i6ti tov aTcb ZA. iibl^(ov &Qa ij AE sid^Bta tfjg ZA 
edd^Bvag. Tcal tityLritai ij A SC%a r^ AK' iAv dh tQc- 
y&vov ii yaovia 8C%a t^ri%^^ t& tfjg pd^scog tfirliiata 
tbv aiftbv B%Bi X6yov talg tov tQLy6vov xXBVQatg' 6 
(ibC^cov &Qa Tcal ij EK tfig KZ, i&v Sh xal dQ^ij fl 
afi^XBta Btri i] Z^ aitd^BV i] aTtdSBc^tg' iv tQLyd^vp 
yaQ tp AZE dgd^^v tJ a^^lBtav i%ovtv tijv Z [ibC^cov 
i6tac ii AE t^g AZ. xal titiirjtaL i] A SC%a t^ AK^ 
i&v Sh tQvy^vov ycovCa SC%a rfMj^ Tcal ta i|ijff' liBC^an; XO 
&Qa xal ii EK tf]g KZ. 

27. Kal aii^lBtd i6tiv p. 318, 11] iuBiSii ii aTcb 
tov xivtQOv ijtl tijv atpiiv iTeL^Bvyvviiivri dQdijv jcovst 
tijv i)7cb ®AK^ d^sta &Qa ij i)7tb ®KA* diipXsta &Qa 

^ iTtb AKE. 16 

28. Oixovv aTtb tov xvQtov p. 318, 19] idv ydQ tb 
xivtQOV Xafidvtsg tfjg 6(paCQag cbr' aitov im^sii^cjfisv 
ijtl tb 6q6^bvov xal ixf^dkcoiisy 6}g iv totg TtQb avtov^ 
^BcoQrjd^^estav tb EA iv ycovCa ty V7tb ABF^ &6%bq 
xal iv totg %Qoka^ov6iv* td ydQ bQ&^sva itdvta iv 20 
ycovCcf bQdtai. 

29. "ChtSQ dSvvatov p. 320, 10] iTtsl yaQ i^ TtQbg 
tp I t6ri i6tl tfl VTtb XAE Sid tijv dvdxka6vv^ ii 
^Qbg ta M &Qa iXd66G)v i6tlv tfjg TtQbg tdi I* Ttokkp 
^kiov &Qa tfjg ixb FAB. 26 



27. V(q). 28. V(q). 29. V(q euan.). 



1. t6] td V. 15. AKE] V, KAE p. 23. /] e corr. V. 
XAE] X ponendum in parte sinistra speculi. 



356 SCHOLIA IN CATOFTRICA. 

30. ^O ^vXkoyi^y^bg oiitfo' t6Se tovSs Ska66ov' tdSe 
taiSs t6ov' tdSs &Qa tovSs iXa66ov. 

31. ^^xa di) tBfut p. 322, 13] i&v y&Q i^pamo- 
{livriv iydycofiev Stit tov F, at y,lv 'bitb r^? itpccjcto- 

6 liivrjg xal f^g ®K yivdiievat C6ar dgd^ yccQ exatiqa' 
i^ mv at &7CoXaiiPav6iievat, inb t&v iq>amofiev(Dv xal 
t&v &vaxkc3(iiv(DV t6at Sl& tb t&g avaxktofiivag t6ag 
elvat^ il^ &v t&g xeQatoevSelg t6ag SlA tb 6%6Xiov tb 
iv tp a' [8]. Xovjt^ aQa ij iTcb BE^^) SCxa ti^vetai. 

10 32. 'E<yrt Sh xal ikd66cov p. 322, 20] iTceiS'^ ^ fihv 
iTcb BFK t6ri trj ijcb ©FE' xard: xoQV(piiv ydff ij Sl 
iTcb ®rE iXd66c}v r^g ixtbg tgiyAvov tov GFE. 

33. *H a-dri} p. 324, 12] tovte6tvv fi &%b tr\g ^et- 
tovog 6q>aCQag' Svvatbv ydcQ xal xatct TcXeCovag axttvag 

16 6qccv. 

34. Tovto Sh iTcdvco p. 324, 15] iv aitp aqa t& 
d^ecoQrjiiatL a%b tcbv SLxotofiL&v t&v ycovi&v. 

35. Ovxovv tcov iiipecjv iniyi^taL p. 326, 3] Sia 
tb tov y' ^i^XCov tfjg iitvTciSov [8]* fi iXa%C6tri yccQ ff 

20 ^etai^i) tov te ^rj^ieCov xal trjg ScafiitQOv^ t&v Sh TCQog 
tijv xvQtijv %eQLq>iQeiav del 'fj iyyLOV tfig ^eta^v tov 
te 6rj^eCov xal trjg SLa^iitQOv trjg dnd)teQOV iXdttajv. 

1) Debuit dici BFJ; sed in BEz/ similiter ratiocinandum 
est, et fortasse huius rei mentio excidit. 



30. Vqi (ad p. 320, 11 sq.). 31. V(q). 32. V(q^). 

33. Vq^ 34. Vq^ 36. V^q^. 



11. GFE] E e corr. V. ^OQvtpi^v] xopvqp^ V. Ad 

p. 324, 20 in V adscribitur: mg •ncctcc tt)v •iioctayQa(pi}v xriv iyjto- 
nsi^iivTiv. 19. T(J] om. q. pipXiov] om. q. Post iitiTiiSov 
^ ^upra scr. Evv,Ui8ov m. rec. V, m. 1 q^. 



SCHOLIA IN CATOPTRICA. 357 

36. Odxovv tffri i^ J5 p. 826, 12] n;a6aL yd^Q at &jch 
xov TcivxQOv t6ag 7Coiov6c ycoviag xatA tiiv i^paQinoyiiv 
xa>v fiiiLXVxXicov. 

37. Obxovv ineClfov itfXLV p. 826, 22] &g iiBi^ovog 
xiii^^axog oi6a xaxa xb ky' xov y' ^i^Xiov xilg iia- 6 
iciSov. 

38. ^Avaxlio^Bvav a[ Hil^Big p. 828, 21] ii^v dnb xov K 
im^B^il^aifiBv inl tb xivxQOV^ tovti^tb xb Z, i6ovxat at 
x&v fiinLXvxkCiDV t6at xatic xijv i(paQ[ioyiiv i^ ixb ^KZ 
xfi inb ZKA. &6tB ij inb ^K® ikdtxcov tfig inb IC 
ZKAy noXX& Ttkiov xrlg 'b%b 0KA. 6y,oCa)g xal iav 
aitb xov N iiat,B'6^G)iLBV inl tb Z. &6xb avaxkibiiBvav 
at bi^Big at ®K^ MN Vilov6iv hg al KA^ NS Sca xb b\ 

39. ^AvaxXoybivri ijlc^ p. 330, 10] inBl yctQ diio at 
BZr Sv6lv xatg EZF t6ai xal ytovCa yatvCa^ Tcal U 
Ttdvxa 7ca6tv' &6xs Tcal i^ i)7cb BFZ t6xi xfi iTcb ZFE. 
i^CBl oiv SXtj ii xov iiinixvxXCov i)7cb AFZ SXy xtj inb 
ZFA t6% 4 ^v ii i)%b BrZ t^ri xy imb ZTE^ loiTcii 
&Qa ii i)%b AFB koLTCfl xfi i)nb EF^ t^rj. i^^Bi &Qa 

il Br tiifLg iTcl xb E. 2( 

40. jdlCxa &Qa xiiLri%^6Bxav p. 830, 18] inBl y&Q t6rj 
il xov ij^LXvxkCov xfj xov ijiiixvxXCov ^ i^ i)v at i>7cb 
B®Aj jd®E t6ai Sia xb tcq&xov^ SC%a &Qa xixiirixai. 

41. Kal dvdXoyov i6tac p. 330, 14] <J*<J; tb y' 
tov ff' Pi^kCov trig iTCtJciSov. ^l 

36. V(qO. 37. V(q»). 38. V(q). 39. V(q). 40. V(q). 
41. Vq\ 



4. flbg] ag iv V. 11. 0KA] &Kd V. 16. BTZ] B 

supra scr. m. 1 V. ^^^ ZT V. 17. imi^vaXiov] i^- 

nvtiXiov V. 24. y'] q, t' V. 



358 SCHOLIA IN CATOPTRICA. 

42. ^icc rovto \lC(x, ii6vri^ i%Bi8i^^ el fjv koI &XXrij 
slxbg ^v aiftag 6vii7ts(f6tv. 

43. Qd ydcQ 6v^%66elxui p. 330, 19] iTtsiSii navrog 
tQLy(bvov a[ ^ ycDvCai Svo 6q^wv ikdttovg, 

6 44. Kal iTCel iisC^cov i6tCv ii BF ^. 332, 17] ijcel 
yccQ il rn t^ri tfj IIK^ ii FN iieC^cav tijg NK. &6te 
xal tb &7tb tov. xoLvbv 7CQ06x6C6^Gi tb &7tb NB* t& 
ccQa aTtb t&v anb (leC^ova. akk& totg (ihv ojtb FNB 
t6ov tb aTtb rBytotg dh ccnb BNK l6ov ro &7tb BK- 
10 &6t6 ii rB fieC^cov rrig BK. 

45. "il6te xal i] ijtb FB® ^ieC^fov p. 332, 18] ijtel 
y&Q tQiyiovov rov FBN at y ycovCav tatg tQi6lv yo- 
vCaig tQLytovov tov BNK t6ai^ e^ mv at <Jt5o ii TtQbg 
tp P xal 'fi iTtb BNK iieC^ovg t&v Svo trjg te TtQbg 

15 rc5 I xal trjg 'bitb FNB^ XoLTtij ccQa i] iTtb KBN Xoi- 
Ttrjg trjg iTtb FBN iXa66G)v' STtov y&Q tb fiet^ov^ iicet 
tb iXattov. 

46. Tovti6tL tfig i)7tb BSK p. 332, 19] t6ri y&Q 
il BK tfj K®, iTteLSij Svo al BFK Sv6lv tatg @rK 

20 t6aL xal ycovCa yG)vCa. 

47. Ovx aQu 6vii7te66ttaL p. 332, 19] i^tel y&Q 
^6C^G)v ii V7tb FBK tfjg i)7tb B®K^ xoLvii TtQO^xeC^d^o} 
ij i)7tb ®BO' al Stio ccQa t&v S^o fieC^ovg. «AA' at 
Svo Sv6lv dQd^atg t6aL' at Svo ccQa Svo dQd^&v ikat- 

25 tovg. &6t6 a[ i)%b FB®^ T®B Sijo dQd^&v ^eC^ovg. 



42. V(qO; adp. 330, 17. 43. V(q^). 44. V(q). 45. 
V(q). 46. V(q^). 47. V(q). 



7. rov] sc. rov dcno, 23. 0B*] B0*, B e corr. 

m, 1, V. 



SCHOLIA IN CATOPTRICA. 359 

48. Msitfov ifftlv ff BZ f. 334, 15] dtd t&v &no 
Sg iv rp ke'. 

49. 2}v^7ts6ovvtat &Qa p. 334, 17] xoivrlg tcqo^" 
X€L^ivrjg r^g inb ZATI. 

50. IlaQAXXriXov yccQ si^iv p. 334, 22] i^jtsl yaQ 5 
dvo aC BAZ Sv6lv talg TAZ l6ai^ &.XXk xal yfovCa 
161(1' 6Qd"^ yicQ sxatiQa' xal ycovia rj i)7tb ABZ ycovia 
tfj i)7cb ArZ 1671. i6tiv 8% xal i] {)7cb ABA t6ri r^ 
vnb AFA' &6ts koiTt^ fj iTtb IIBA XoLTtfi tfj {)7tb 
@rA t6ri Sl& tb titQa6iv d^d^atg l'6ag slvai tag i)7tb 10 
UAB t6rj tfj xatcL xoQvq^^^v. i&v Sh Svo tQiycjva Siio 
ycavCag Svo yavCavg t6ag ixrj xal ta s^rlg' t6ov &Qa 
tb BAU tQCyavov t& ©AF tQiy(ovG). xoivbv 7tQo6- 
xsC6^G) tb BAT' tb 11 TB &Qa rp ©BT t6ov. xaC 
si6iv iTtl tfjg ai)tfig pd6scog trjg 11®' 7taQakkriXog &Qa 15 

il Br tfj n@. 

51. MsC^ov y&Q i] MA p. 336, 2] iTtsl i6oy6vi6v 
i6tL ro UAM tQCycovov ta ABA tQLycavc)' at ^lv 
dQd^al ai)tcjv t6aL^ rj S^k 7tQbg tp K tfj V7tb UAM 
xata ta ^Srj SsL%^ivta' XoLTtij &Qa tfi koLTtfi t6rj' tcbv 20 
S% Idoycovicav avdkoyov al TtXsvQal at 7tSQl tag t6ag 
ycoviag^ i6tLv &Qa^ ijg ij IIM TtQbg MA^ ovtcog ij BA 
TtQbg AA. xal ivakXd^^ &g ij 11 M TtQbg BA^ oiitcog 

ij MA TtQbg AA. (isi^ov S^ ij IIM t^g BA' iSsix^rj 
ydQ' xal ij MA &Qa rijg AA. 26 

48: Vq^ 49. V. 50. V(q). 51. V(q). 



4. ZAU] Zz/JT V. 7. ABZ] AZB V. 8. ATZ] 

AFH V, 10. trag] hic aliquid excidit ( ^eri 81 xal 17). 

12. ^xvf] ^x^L V. 13. %OLvbv TtpooTislad^m^hiaY. 15. !!€>] 
debuit Br. 18. ABA] ABF Y. 25. MA] MAA V. xfis 
AA] tfj TAA V. 



360 SCHOLIA IN CATOPTRICA. 

52. MaCtfiov 8% fi MA triq AA oiJrog' btEl Ttaq- 
dkkriXog 'fj BA t^ TIM^ t6ri fi Tcgbg tm M ymvia t^ 
TtQog tp A^ iTtstSij dQd^ij ij nQog ro A. i6tt di xal 
'fl TtQog ro K teri r^ iTtb IIAM Svit tb ti^v [ikv XQog 

6 t^ K t6riv Bivav tfj TtQbg rp jd^ tijv d^ TtQbg tp A 
trj xat& xoQvg)7Jv' l6oy6viov &Qa tb IIAM tQiyavov 
ta BAA, t&v 8\ i6oy(ovi<ov avAXoyov at TtX^QoX al 
nsql td^g tcag ycoviag' i6tvv &Qa^ d)g ii UM TtQbg tijv 
MAj ovtfog ij BA itQbg AA' xal ivakkd^^ i}g ij 11 M 

10 TtQbg tijv BA^ ovtcog ij MA TtQbg AA, iisi^c^v 8h ij 
MA tfjg AA' iiLsitjtov &Qa xal i^ UM tr^g BA^) b^icog 
xal ii M& tfjg AR 

53. Ovxovv tb ^lv B p. 336, 12] i%sv8ii sxa6tov 
t&v 6qcoiisvcov bQatai xatcc ti^v Cii^TttcD^iv ixfiakko- 

15 iisvG)v trig ts ^ipscjg xal tj\g aitb tov bQca^svov inl 
tb xsvtQOV iTtc^svywiisvrjgj &6ts tov B bQ&vtog tov F 
6qo^svov xal tov F bQ&vtog tov B 6qcoiisvov tavta 
yivs^d^ai. 

54. ^EXd66G)v Sh r] EK tf^g BF ix tov i6oyc)VLOv 
20 slvaL tb ^AF r© AEK^) ix trjg xoLvrjg ycoviag xal 

ix tcbv dQ^-cbv Sia ro TtaQaXkijkovg slvai tijv EK xal 
tijv BT, 

1) Errore permutauit MA, AA et JTM, BA. 

2) Debuit dici: dimidium JEK. 



52. V(q); eodem pertinet, 53. V(q). 54. V(q); pertinet 
ad p. 336, 14 sq. 



4. Tc5] corr. ex rd m. 1 V. 6. rfj] r^g V. xopvqpijv] 

TiOQVtpfigY. IIAM] nAN Y. 9. MA] corr. ex MA 

m. 2 V. 10. MA] MA V._ 11. MA] MA V. 16. rov B] 
rb B Y. 17. Tavrcc] ravra V. 



SCHOLIA IN CATOPTRICA. 361 

55. UaQalXrikog yccQ i^rcv 'fj EK p. 336, 15] TtdXtv 
6(wi(og l6oyG}viov dsLxwiievov xov KZF tQiyfhvov tm 
EZB tQiyAvG) Tcal fLi&g TtksvQ&g /Ltta TCksvQa ttfrjg tr^g 
nffhg tatg C^avg ycoviaig. 

56. 'H yaQ ycjvia ff JtQog tfj p. 340, 9] ictv y&Q i 
&rtS£'i$S(Dft£i/ ti^v &7tb tov r iitl ro 0, C6ri ictlv fi iTtb 
HFG tjH imb ©FB' 'fiiiixvxXicov yaQ. oixovv i^ vTtb 
HRd iXd66(DV tijg ^Ttb ®rB' nokXm Ttkiov tijg ijtb 
2MB. 

^ia ti Sh ij &vaxkG)^ivrj fw) ijtl tb xivtQOv iTti- U 
i^^dywtai; iTteiSii at HifeLg iv t^avg ycoviaig &va- 
xX&vtav^ iXdttoyv S% iyLsXkBV alvai i] TtQbg t^ U tf^g 
'bnb SFB^ ivdyxrj ohv f^v C6rjv tf] JtQbg t& 11 aitb 
tfig (isi^ovog &q)aLQsd'6t6av tf^g iTtb ©FB ivcotiQG) nov 
jeotfj6aL tiiv &vAxXa6iv hg iitl tb K n 

57. OavBQbv oiv^ Srt p. 340, 13] iav iTtL^sii^G^iisv 
iatb tov inl tb F xal inl tb A^ i6tai Sflkov oiitoag' 
iTtsl di5o at K®r Sv6lv talg KSA t6av xal ycovia 
ycDvia Sl& t&g nsQi^psQsiag^ Ttdrrta Jta6iv' ci6ts ycovia 

il inb KA® yG)via tfi i)nb KFS t6rj. Ttdhv ijtsl ij 2 
imb AQjd t6ri rg i)nb FQA Sia tb tag ^ri^si6ag t6ag 
yoDviag ix t&v ts66dQG)v dQd^cjv tag vitoXoiitovg Siio 
t6ag xatakiiiTtdvsLVj Svo ai r®A Sv6lv tatg A0A 
t6ai xal ycovia ycjvia' t6ri aQa i] vTtb SA^ r/J i)Jtb 
®rA. insl ovv Uri ij i)%b &AB t^rj tfj vitb &rB' 'i 
illiixvxkicov y&Q iq>aQfLO^oiiivG)v' i^ cjv ij vitb KA0 
t6ri ty i>7tb KF®^ koLitii ccQa ii iitb KAB t6ri tfj vTtb 
KFB, &n' ii i)7tb KTB t6r] ty iitb AnH' ^ aQa 



66. V(q^). 56. V(q). 57. V(q). 

HvolidoBf edd. Hoiberg ot NeivgCK. "VW. 'LV^'^ 



362 SCHOLIA IN CATOFrRICA. 

i}7eb JTH t6ri xfi i)jth KAB. akX' ii ino /tTH t6ri 
tfi imb JAP' xal fi i)%b KAB &(fa rfi imb AAP. 

58. ^Enl t&Q SlA tov xevtQov p. 342, 7] tovti6ti 
xatdi tf^g B® 7ta6ttt TtQog aXXtp xal &kk(p 6rj^ECai^ ivl 
5 d\ sxatiQcad^ev &6nsQ aC FKA. 

68. V(q). 



4. &XXip (pr.)] q, &XXo V. 



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